Chapter 14.5 - Miscellaneous Exercise

In the previous section, we saw validating statements. In this section, we will see some miscellaneous examples.

Solved example 14.14
Check whether “Or” used in the following compound statement is exclusive or inclusive. Write the component statements of the compound statement and use them to check whether the compound statement is true or not. Justify your answer.
t: you are wet when it rains or you are in a river.
Solution:
1. We have:
t: you are wet when it rains or you are in a river.
2. The component statements are:
p: you are wet when it rains.
q: you are wet when you are in a river.
   ♦ p and q are connected by “or”.
3. Let us analyze the component statements:
• During rain, you are wet.
• When in a river, you are wet.
• When in a river, if it rains, you are wet.
• So the “or” used in this case is inclusive “or”.
4. Both the component statements are true. So t is true.

Solved example 14.15
Write the negation of the following statements
(i) p: For every real number x, x² > x.
(ii) q: There exists a rational number x such that x² = 2.
(iii) r: All birds have wings.
(iv) s: All students study mathematics at the elementary level.
Solution:
Part (i)
1. Consider the statement:
p: For every real number x, x² > x.
2. We want ~p.
• p tells us that: For all real numbers, the given property is valid.
• ~p must tell us that "For all" is false.
3. This ~p can be achieved by saying that there is at least one real number for which the property is not valid.
• So we get:
~p: There exists one real number x for which x² ≤ x.

Part (ii)
1. Consider the statement:
q: There exists a rational number x such that x² = 2.
2. We want ~q.
• q tells us that: There is at least one rational number for which the given property is valid.
• ~q must tell us that "at least one" is false.
3. This ~q can be achieved by saying that for all rational numbers, the property is not valid.
• So we get:
~p: For all rational numbers x, x² ≠2.

Part (iii)
1. Consider the statement:
r: All birds have wings.
2. We want ~r.
• r tells us that: For all birds, the given property is valid.
• ~r must tell us that "For all" is false.
3. This ~r can be achieved by saying that there is at least one bird for which the property is not valid.
• So we get:
~r: There exists one bird which do not have wings.

Part (iv)
1. Consider the statement:
s: All students study mathematics at the elementary level.
2. We want ~s.
• s tells us that: For all students, the given property is valid.
• ~s must tell us that "For all" is false.
3. This ~r can be achieved by saying that there is at least one student for which the property is not valid.
• So we get:
~r: There exists one student who does not study mathematics at the elementary level.

Solved example 14.16
Using the words “necessary and sufficient”, rewrite the statement “The integer n is odd if and only if n² is odd”. Also check whether the statement is true.
Solution:
Part (i): Rewriting the statement
Using the words “necessary and sufficient”, we can write:
“The integer n is odd” is necessary and sufficient condition for “n² to be odd” and vice versa. 

Part (ii): Checking the validity.
• The given compound statement is in the form “p if and only if q”.
The component statements are:
p: The integer n is odd.
q: n² is odd.
• In such cases, we know that:
    ♦ Whenever p is true, q is also true.
    ♦ Whenever q is true, p is also true.
• So we need to show two items:
Case (i) If p is true, then q is true.
Case (ii) If q is true, then p is true.

Case (i): If p is true, then q is true.
    ♦ First, we assume that p is true.
    ♦ Based on this assumption, we check q.
1. Assuming p to be true:
• Let us assume that, integer n is indeed odd.
2. Since it is an odd integer, we can write:
    ♦ n = 2m + 1
        ✰ Where m is some integer.
3. Now we calculate n². We get:
$\begin{array}{ll}
{}&{n^2} & {~=~}& {(2m+1)^2} &{} \\
{}&{} & {~=~}& {4m^2 + 4m + 1} &{} \\
{}&{} & {~=~}& {2(2m^2 + 2m) + 1} &{} \\
\end{array}$

4. Consider the result in (3):
• m is an integer.
⇒ m² will be an integer.
⇒ (2m² + 2m) will be even.
⇒ 2(2m² + 2m) will be even.
⇒ 2(2m² + 2m) + 1 will be odd.

5. Based on (3) and (4), we can write:
n² is odd.
• So the statement q is true.
6. We obtained the truth value of q as T.
• We obtained this by assuming that the truth value of p is T.
• So the truth value of “if p then q” is T.

Case (ii): If q is true, then p is true.
Here we will use the contrapositive method. That is., if ~p then ~q.
    ♦ First, we assume that ~p is true.
    ♦ Based on this assumption, we check ~q.
The statements are:
~p: The integer n is even.
~q: n² is even.
1. Assuming ~p to be true:
• Let us assume that, n is an even integer and n² is indeed even.
2. Since it is an even integer, we can write:
    ♦ n = 2m
        ✰ Where m is some integer.
3. Now we calculate n². We get:
$\begin{array}{ll}
{}&{n^2} & {~=~}& {(2m)^2} &{} \\
{}&{} & {~=~}& {4m^2} &{} \\
{}&{} & {~=~}& {2(2m^2)} &{} \\
  \end{array}$

4. Consider the result in (3):
• m is an integer.
⇒ m² will be an integer.
⇒ 2m² will be even.
⇒ 2(2m²) will also be even.

5. Based on (3) and (4), we can write:
n² is even.
• So the statement ~q is true.
6. We obtained the truth value of ~q as T.
• We obtained this by assuming that the truth value of ~p is T.
• So the truth value of “if ~p then ~q” is T.
• So the truth value of “if q then p” is T.

◼ From case (i), we obtained:
• The truth value of “if p then q” is T.
◼ From case (ii), we obtained:
• The truth value of “if q then p” is T.
◼ So we can write: The given "p if and only if q" is true.
• That is.,
The statement “The integer n is odd if and only if n² is odd” is true.

Solved example 14.17
For the given statement, identify the necessary and sufficient conditions
t: If you drive over 80 km per hour, then you will get a fine. 
Solution:
1. Consider the statement:
t: If you drive over 80 km per hour, then you will get a fine.
2. The component statements are:
p: You drive over 80 km per hour.
q: You will get a fine.
3. This is a statement with "if p then q".
• So we can write:
   ♦ p is a sufficient condition for q.
   ♦ q is a necessary condition for p.
(see example 2 at the beginning of section 14.3)
4. Thus we get:
• The sufficient condition is:
Driving over 80 km per hour.
(This is sufficient to get a fine)
• The necessary condition is:
Getting a fine.
(This is necessary if the speed is above 80 km per hour)

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The link below gives some more solved examples:

Miscellaneous Exercise

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In the next chapter, we will see statistics.

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Chapter 14.3 - Contrapositive And Converse

In the previous section, we saw quantifiers. In this section, we will see Implications.

Implications

• In mathematics, we often encounter statements with:
    ♦ “if-then”
    ♦ “only if”
    ♦ “if and only if”
Let us see some examples:
Example 1
This can be written in 4 steps:
1. Consider the following statement:
r: If you are born in some country, then you are a citizen of that country.
2. We can see that, r is a compound statement. It’s component statements are:
p: You are born in some country.
q: You are citizen of that country.
3. Now r becomes: if p then q
• “if p then q” has a definite meaning.
We can write:
    ♦ Whenever p is true,
    ♦ q is also true.
• We can also write:
    ♦ Whenever p is false,
    ♦ We must discard the compound statement r.
4. An important point to remember is:
“if p then q” does not give any guarantee that p happens.

Example 2
This can be written in 4 steps:
1. Consider the following statement:
r: If a number is a multiple of 9, then it is a multiple of 3.
2. We can see that, r is a compound statement. It’s component statements are:
p: A number is a multiple of 9.
q: That number is a multiple of 3.
3. Now r becomes: if p then q
• “if p then q” has a definite meaning.
We can write:
    ♦ Whenever p is true,
    ♦ q is also true.
• For our present example:
    ♦ Whenever a number is a multiple of 9
    ♦ That number will be a multiple of 3 also.
• We can also write:
    ♦ Whenever p is false,
    ♦ We do not know whether q is true or false.
• For our present example:
    ♦ Whenever a number is not a multiple of 9
    ♦ We must discard the compound statement r.
4. An important point to remember is:
“if p then q” does not give any guarantee that p happens.
• For our present example:
Just by reading r, we cannot say that a number is a multiple of 9.
• r is a link between p and q.
• p cannot stand alone independently.

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Based on example 2, we can write some more details. It can be written in 5 steps:
1. When we are given “if p then q”, we can write it in another form:
"p implies q"
• Symbolically, we denote this as: p ⇒ q
    ♦ The symbol "⇒" stands for “implies”.
• For our present example, we can write:
    ♦ A number is a multiple of 9.
    ♦ implies that
    ♦ That number is a multiple of 3.
2. When we are given “if p then q”, we can write:
"p is a sufficient condition for q".
• For our present example, we can write:
    ♦ Knowing a number to be a multiple of 9,
    ♦ is sufficient
    ♦ To know that, the number is a multiple of 3.
3. When we are given “if p then q”, we can write:
"p only if q".
• For our present example, we can write:
    ♦ A number is a multiple of 9
    ♦ only if
    ♦ That number is a multiple of 3.
4. When we are given “if p then q”, we can write:
"q is a necessary condition for p"
• For our present example, we can write:
    ♦ For a number to be a multiple of 9
    ♦ it is necessary that
    ♦ That number is a multiple of 3   
5. When we are given “if p then q”, we can write:
"~q implies ~p"
• For our present example, we can write:
    ♦ When a number is not a multiple of 3
    ♦ it implies that
    ♦ That number is not a multiple of 9.   
◼ When we are given “if p then q”, we can rewrite it in the above five different ways.

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Contrapositive statement

• When we are given “if p then q”, we can write:
“if ~q then ~p”.
• This is known as the contrapositive of “if p then q”.
• Note that, the order of p and q is changed in the contrapositive statement.
• Let us see some examples:

Example 1:
This can be written in 4 steps:
1. Consider the statement:
r: If a number is divisible by 9, then it is divisible by 3.
2. The component statements are:
p: A number is divisible by 9.
q: That number is divisible by 3.
3. Let us write the negation statements:
~p: A number is not divisible by 9.
~q: That number is not divisible by 3.
4. Now we can write “if ~q then ~p”:
If a number is not divisible by 3 then that number is not divisible by 9.
• This is the contrapositive of r.

Example 2:
This can be written in 4 steps:
1. Consider the statement:
r: If you are born in India, then you are a citizen of India.
2. The component statements are:
p: You are born in India.
q: You are a citizen of India.
3. Let us write the negation statements:
~p: You are not born in India.
~q: You are not a citizen of India.
4. Now we can write “if ~q then ~p”:
If you are not a citizen of India, then you are not born in India.
• This is the contrapositive of r.

Example 3:
This can be written in 4 steps:
1. Consider the statement:
r: If a triangle is equilateral, it is isosceles.
2. The component statements are:
p: A triangle is equilateral.
q: That triangle is isosceles.
3. Let us write the negation statements:
~p: A triangle is not equilateral.
~q: That triangle is not isosceles.
4. Now we can write “if ~q then ~p”:
If a triangle is not isosceles, then that triangle is not equilateral.
• This is the contrapositive of r.

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Converse statement

• When we are given “if p then q”, we can write:
“if q then p”.
• This is known as the converse of “if p then q”.
• Note that, the order of p and q is changed in the converse statement.
• Let us see some examples:

Example 1:
This can be written in 3 steps:
1. Consider the statement:
r: If a number is divisible by 10, then it is divisible by 5.
2. The component statements are:
p: A number is divisible by 10.
q: That number is divisible by 5.
3. Now we can write “if q then p”:
If a number is divisible by 5, then that number is divisible by 10.
• This is the converse of r.

Example 2:
This can be written in 3 steps:
1. Consider the statement:
r: If a number n is even, then n² is even.
2. The component statements are:
p: A number n is even.
q: The square of that number n² is even.
3. Now we can write “if q then p”:
If the number n² is even, then n is even.
• This is the converse of r.

Example 3:
This can be written in 3 steps:
1. Consider the statement:
r: If you do all the exercises in the book, then you get an A grade in the class.
2. The component statements are:
p: You do all the exercises in the book.
q: You get an A grade in the class.
3. Now we can write “if q then p”:
If you get an A grade in the class, then you have done all the exercises in the book.
• This is the converse of r.

Example 4:
This can be written in 3 steps:
1. Consider the statement:
r: If two integers a and b are such that a > b, then (a-b) is always a positive integer.
2. The component statements are:
p: Two integers a and b are such that a > b.
q: (a-b) is always a positive integer.
3. Now we can write “if q then p”:
If two integers a and b are such that (a-b) is always a positive integer, then a > b.
• This is the converse of r.

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Let us see a solved example:

Solved example 14.8
For each of the following compound statements, first identify the corresponding component statements. Then check whether the statements are true or not.
(i) If a triangle ABC is equilateral, then it is isosceles.
(ii) If a and b are integers, then ab is a rational number.
Solution:
Part (i):
1. Consider the statement:
r: If a triangle ABC is equilateral, then it is isosceles.
2. The component statements are:
p: A triangle ABC is equilateral.
q: That triangle is isosceles.
3. An equilateral triangle has all three sides equal. So obviously, the two sides other than the base will be equal. That means, an equilateral triangle is an isosceles triangle also. 
• We can write:
If p happens, q is true. So r is true.

Part (ii):
1. Consider the statement:
r: If a and b are integers, then ab is a rational number.
2. The component statements are:
p: a and b are integers.
q: ab is a rational number.
3. Integers do not have any decimal portions. So the product of two integers will not have any decimal portions. That means, the product will be an integer. An integer is a rational number.
• We can write:
If p happens, q is true. So r is true.

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“If and only if” statement 

This can be explained in steps:
1. In some cases, we encounter the condition: “p if and only if q”
2. Let us see an example:
• A student can apply for MSc Maths if he has taken Maths, Physics and Computer science for the Bsc course.
• A student can apply for MSc Chemistry if and only if he has taken Maths, Physics and Chemistry for the Bsc course.
3. In some other cases, both the conditions below need to be satisfied:
    ♦ “p if and only if q”
    ♦ “q if and only if p”
• In such cases we use the symbol "⇔"
    ♦ Using this symbol, we can write: p ⇔ q.
4. Note that:
(i) “p if and only if q”
    ♦ is different from
    ♦ p ⇔ q.
(ii) “q if and only if p”
    ♦ is different from
    ♦ p ⇔ q.
(iii) “p if and only if q”
    ♦ is different from
    ♦ “q if and only if p”
5. If we are given "p ⇔ q", then all the four conditions below must be satisfied:
(i) p if and only if q
(ii) q if and only if p
(iii) p is necessary and sufficient condition for q
(iv) q is necessary and sufficient condition for p

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Let us see a solved example:
Solved example 14.9
Given below are pairs of statements. Combine them using “if and only if”.
(i) p: If a rectangle is a square, then all it’s four sides are equal.
q: If all the four sides of a rectangle are equal, then that rectangle is a square.

(ii) p: If the sum of digits of a number is divisible by 3, then the number is divisible by 3.
q: If a number is divisible by 3, then the sum of it’s digits is divisible by 3.

(iii) p: If a tumbler is half empty, then it is half full.
q: If a tumbler is half full, then it is half empty.

Solution:
Part (i)
1. Consider the statement:
p: If a rectangle is a square, then all it’s four sides are equal.
• This p can be considered as a "if r then s" statement.
• So the component statements are:
    ♦ r: A rectangle is a square.
    ♦ s: All four sides of that rectangle are equal.
• It is clear that "if r then s" is applicable.
2. Consider the statement:
q: If all the four sides of a rectangle are equal, then that rectangle is a square.
• This q contains r and s that we saw above. It can be considered as a "if s then r" statement.
• So the component statements are:
    ♦ s: All the four sides of a rectangle are equal.
    ♦ r: That rectangle is a square.
• It is clear that "if s then r" is applicable.
3. So both the two conditions below are applicable:
    ♦ "if r then s"
    ♦ "if s then r"
4. Since both the conditions are applicable, we can combine r and s using "if and only if". We get:
A rectangle is a square if and only if all four sides of that rectangle are equal.

Part (ii)
1. Consider the statement:
p: If the sum of digits of a number is divisible by 3, then the number is divisible by 3.
• This p can be considered as a "if r then s" statement.
• So the component statements are:
    ♦ r: The sum of digits of a number is divisible by 3.
    ♦ s: The number is divisible by 3.
• It is clear that "if r then s" is applicable.
2. Consider the statement:
q: If a number is divisible by 3, then the sum of it’s digits is divisible by 3.
• This q contains r and s that we saw above. It can be considered as a "if s then r" statement.
• So the component statements are:
    ♦ s: A number is divisible by 3.
    ♦ r: The sum of it’s digits is divisible by 3.
• It is clear that "if s then r" is applicable.
3. So both the two conditions below are applicable:
    ♦ "if r then s"
    ♦ "if s then r"
4. Since both the conditions are applicable, we can combine r and s using "if and only if". We get:
The sum of digits of a number is divisible by 3 if and only if the number is divisible by 3.

Part (iii)
1. Consider the statement:
p: If a tumbler is half empty, then it is half full.
• This p can be considered as a "if r then s" statement.
• So the component statements are:
    ♦ r:  A tumbler is half empty.
    ♦ s: That tumbler is half full.
• It is clear that "if r then s" is applicable.
2. Consider the statement:
q: If a tumbler is half full, then it is half empty.
• This q contains r and s that we saw above. It can be considered as a "if s then r" statement.
• So the component statements are:
    ♦ s: A tumbler is half full.
    ♦ r: That tumbler is half empty.
• It is clear that "if s then r" is applicable.
3. So both the two conditions below are applicable:
    ♦ "if r then s"
    ♦ "if s then r"
4. Since both the conditions are applicable, we can combine r and s using "if and only if". We get:
A tumbler is half empty if and only if that tumbler is half full.

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The link to some solved examples is given below:

Exercise 14.4

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In the next section, we will see validating statements.

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Chapter 14.2 - Special Words And Quantifiers

In the previous section, we saw compound statements and component statements. In this section, we will see Special words/phrases.

Special words/phrases

• We saw that, component statements are connected by words like “and”, “or” etc.,
• These words are known as connectives.
• When used in mathematical statements, these words have special meanings. We need to clearly understand those special meanings.

The word “And”

This can be explained in 3 steps:
1. First we will see an example. It can be written in 4 steps:
(i) Given below, is a compound statement with the word “and”:
p: A point occupies a position and it’s location can be determined.
(ii) The component statements are:
q: A point occupies a position.
r: It’s location can be determined.
   ♦ q and r are connected by “and”.
(iii) Checking component statements:
• q is true.
• r is true.
(iv) Since both q and r are true, p is true.
2. Let us see another example. It can be written in 4 steps:
(i) Given below, is a compound statement with the word “and”:
p: 42 is divisible by 5, 6 and 7.
(ii) The component statements are:
q: 42 is divisible by 5.
r:  42 is divisible by 6.
s:  42 is divisible by 7.
   ♦ q, r and s are connected by “and”.
(iii) Checking component statements:
• q is false.
• r is true.
• s is true
(iv) Since q is false, p is false.
(Even though r and s are true, p is false because one component q is false)
3. We can write two rules related to “and”:
(i) A compound statement with “and” is true if all it’s component statements are true.
(ii) A compound statement with “and” is false if one or more of it’s component statements is false.

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Let us see a solved example
Solved example 14.5
Write the component statements of the following compound statements and check whether the compound statements are true or false.
(i) A line is straight and extends indefinitely in both directions.
(ii) 0 is less than every positive integer and every negative integer.
(iii) All living things have two legs and two eyes.
Solution:
Part (i):
1. We have:
p: A line is straight and extends indefinitely in both directions.
2. The component statements are:
q: A line is straight.
r: A line extends indefinitely in both directions.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is true
• r is true.
4. Since both q and r are true, p is true.

Part (ii):
1. We have:
p: 0 is less than every positive integer and every negative integer.
2. The component statements are:
q: 0 is less than every positive integer.
r: 0 is less than every negative integer.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is true
• r is false.
4. Since r is false, p is false.

Part (iii):
1. We have:
p: All living things have two legs and two eyes.
2. The component statements are:
q: All living things have two legs.
r: All living things have two eyes.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is false
• r is false.
4. Since one or more of the component statements is false, p is false. In this case both the components q and r are false.

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Let us see an interesting case involving “and”. It can be written in 4 steps:
1. Consider the sentence:
A mixture of alcohol and water can be separated by chemical methods.
2. We are tempted to write two separate sentences:
(i) A mixture of alcohol can be separated by chemical methods.
(ii) A mixture of water can be separated by chemical methods.
3. Let us analyze the two sentences:
(i) The sentence in 2(i) does not have any meaning. This is because, if alcohol alone is present, we cannot call it a ‘mixture’. If there is no ‘mixture’, there is no need for separating.
(ii) The sentence in 2(ii) does not have any meaning. This is because, if water alone is present, we cannot call it a ‘mixture’. If there is no ‘mixture’, there is no need for separating.
4. It is clear that, the word “and” in this case is not used for connecting two statements. It is used for connecting two substances alcohol and water.

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The word “Or”

This can be explained in 6 steps:
1. First we will see an example. It can be written in 4 steps:
(i) Given below, is a compound statement with the word “or”:
p: Two lines in a plane, either intersect at a point or they are parallel.
(ii) The component statements are:
q: Two lines in a plane, intersect at a point.
r: Two lines in a plane, are parallel.
   ♦ q and r are connected by “or”.
(iii) Checking component statements:
• if q is true, then r is false.
• if r is true, then q is false.
(iv) One of the components will be always true.
If one of the components (q or r) is true, p is true.
2. Let us see another example. It can be written in 3 steps:
(i) Given below, is another compound statement with the word “or”:
p: 125 is a multiple of 7 or 8.
(ii) The component statements are:
q: 125 is a multiple of 7.
r: 125 is a multiple of 8.
   ♦ q and r are connected by “or”.
(iii) Checking component statements:
• q is false.
• r is false.
(iv) If all the components are false, p is false.
3. Let us see one more example. It can be written in 3 steps:
(i) Given below, is a compound statement with the word “or”:
p: The school is closed if there is a holiday or sunday.
(ii) The component statements are:
q: The school is closed if there is a holiday.
r: The school is closed if it is sunday.
   ♦ q and r are connected by “or”.
(iii) Checking component statements:
• q is true.
• r is true.
(iv) If one of the components is true, p is true.
4. We can write two rules related to “or”:
(i) A compound statement with “or" is true if one or more of it’s component statements are true.
(ii) A compound statement with “or” is false if all of it’s component statements are false.
5. The word "or" is used in two ways in English language. This can be explained using two examples.

Example 1:
This can be written in 3 steps:
(i) Consider the compound statement with word "or":
p: An ice cream or pepsi is available with lunch in a restaurant.
(ii) The component statements are:
q: An ice cream is available with lunch in a restaurant.
r: A pepsi is available with lunch in a restaurant.
   ♦ q and r are connected by “or”.
(iii) Let us analyze the component statements:
• If a customer does not want pepsi, he can opt for ice cream.
• If a customer does not want ice cream, he can opt for pepsi.
• Only one item (ice cream or pepsi) will be allowed.
• The “or” used in such cases is called exclusive “or”.

Example 2:
This can be written in 3 steps:
(i) Given below, is a compound statement with the word “or”:
p: A student who has taken biology or chemistry can apply for M.Sc. microbiology program.
(ii) The component statements are:
q: A student who has taken biology can apply for M.Sc. microbiology program.
r: A student who has taken chemistry can apply for M.Sc. microbiology program.
   ♦ q and r are connected by “or”.
(iii) Let us analyze the component statements:
• If a student has taken biology, he/she can apply for M.Sc. microbiology program.
• If a student has taken chemistry, he/she can apply for M.Sc. microbiology program.
• If a student has taken both biology and chemistry, he/she can apply for M.Sc. microbiology program.
• The “or” used in such cases is called inclusive “or”.
6. We must clearly understand the difference between exclusive “or” and inclusive “or”. Then only we will be able to check whether a compound statement is true or false.

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Now we will see some solved examples.

Solved example 14.6
For each of the following statements, determine whether an inclusive “or” or exclusive “or” is used. Give reasons for your answer.
(i) To enter a country, you need a passport or a voter registration card.
(ii) The school is closed if it is a holiday or a Sunday.
(iii) Two lines intersect at a point or are parallel.
(iv) Students can take French or Sanskrit as their third language.
Solution:
Part (i):
1. We have:
p: To enter a country, you need a passport or a voter registration card.
2. The component statements are:
q: To enter a country, you need a passport.
r: To enter a country, you need a voter registration card.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• If a person has a passport, he/she can enter a country.
• If a person has a voter registration card, he/she can enter a country.
• If a person has both passport and voter registration card, he/she can enter a country.
• So the “or” used in this case is inclusive “or”.

Part (ii):
1. We have:
p: The school is closed if it is a holiday or a Sunday.
2. The component statements are:
q: The school is closed if it is a holiday.
r: The school is closed if it is a Sunday.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• On a holiday, the school is closed.
• On a Sunday, the school is closed.
• On a day, if Sunday and holiday coincide, then also the school is closed.
• So the “or” used in this case is inclusive “or”.

Part (iii):
1. We have:
p: Two lines intersect at a point or are parallel.
2. The component statements are:
q: Two lines in a plane, intersect at a point.
r: Two lines in a plane, are parallel.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• if q is true, then r is false.
• if r is true, then q is false.
4. Both cannot be true at the same time.
• Both cannot be false at the same time.
• So the “or” used in this case is exclusive “or”.

Part (iv):
1. We have:
p: Students can take French or Sanskrit as their third language.
2. The component statements are:
q: Students can take French as their third language.
r: Students can take Sanskrit as their third language.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• if q is true, then r is false.
• r is true, then q is false.
• No student can take both languages at the same time.
• So the “or” used in this case is exclusive “or”.

Solved example 14.7
For each of the following statements, determine whether an inclusive “or” or exclusive “or” is used. Give reasons for your answer.
(i) √2 is a rational number or an irrational number.
(ii) To enter into a public library children need an identity card from the school or a letter from the school authorities.
(iii) A rectangle is a quadrilateral or a 5-sided polygon.
Solution:
Part (i):
1. We have:
p: √2 is a rational number or an irrational number.
2. The component statements are:
q: √2 is a rational number.
r: √2 is an irrational number.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• if q is true, then r is false.
• if r is true, then q is false.
4. Both cannot be true at the same time.
• Both cannot be false at the same time.
• So the “or” used in this case is exclusive “or”.
5. In this case, r is true. So p is true.

Part (ii):
1. We have:
p: To enter into a public library children need an identity card from the school or a letter from the school authorities.
2. The component statements are:
q: To enter into a public library children need an identity card from the school.
r: To enter into a public library children need a letter from the school authorities.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• If identity card is available, entry is possible.
• If letter is available, entry is possible.
• If both identity card and letter is available, entry is possible.
• So the “or” used in this case is inclusive “or”.

Part (iii):
1. We have:
p: A rectangle is a quadrilateral or a 5-sided polygon.
2. The component statements are:
q: A rectangle is a quadrilateral.
r: A rectangle is a 5-sided polygon.
   ♦ q and r are connected by “or”.
3. Let us analyze the component statements:
• if q is true, then r is false.
• if r is true, then q is false.
(Recall that, a quadrilateral has only four sides)
4. Both cannot be true at the same time.
• Both cannot be false at the same time.
• So the “or” used in this case is exclusive “or”.
5. In this case, q is true. So p is true.

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Quantifiers

Some basics can be written in 4 steps:
1. Consider the set of all rectangles.
• There will be infinite number of rectangles in that set.
2. We know that, squares are also rectangles.
• So some of the members in the set will be squares.
3. This fact can be written as a statement:
p: There exists a rectangle whose all sides are equal.
• This statement means that, there is at least one rectangle whose all sides are equal.
4. The phrase “There exist” is a quantifier.
• Quantifiers are phrases which give us an idea about the “quantity of items”
• “There exists” gives us the idea that, there is at least one item.

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Let us see another quantifier. It can be explained in steps:
1. Let S be the set of all prime numbers.
• Then we can write S in the set-builder form as:
S = {p : p is a prime number}
2. Now, square root of any prime number will be an irrational number.
• We want to write this as a statement. We can write it as:
q: For every prime number p, √p is an irrational number.
3. The phrase “For every” is a quantifier.
• As we mentioned just above, Quantifiers are phrases which give us an idea about the “quantity of items”
• “For every” gives us the idea that, all items in the set are to be taken into consideration.
• In the previous example, we had to consider only some members of the set  (members which are squares).

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It is important to consider the position of the quantifier. This can be explained using an example. It can be written in 5 steps:
1. Consider two sentences:
(i) For every positive number x there exists a positive number y such that y < x
(ii) There exists a positive number y such that for every positive number x, we have y < x
(In both the sentences, “number” means, real number)
2. Let us analyze the first sentence.
• We first consider the set P of positive numbers:
P = {x : x is a +ve number}
• The sentence says that:
We can pick any x from that set. We will be able to find a positive number y such that, y is less than x.
• For example, (y= x/2) will be less than x.
3. Let us analyze the second sentence.
• We first consider the set P of positive numbers:
P = {y : y is a +ve number}
• The sentence says that:
We can pick any y from that set. That y will be less than any positive number x.
• This is not possible. For example, y will be greater than (x= y/2).
• So this sentence is false.
4. Both sentences appear to be the same.
• But we see that:
   ♦ First sentence is true.
   ♦ Second sentence is false.
5. Why is that so?
• In the first sentence, the quantifier “there exists” is at the middle.
• In the second sentence, “there exists” is at the beginning.
• The quantifiers must be carefully introduced in a sentence. Each quantifier must be introduced precisely at the right place. Not too early and not too late.

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The link to some solved examples is given below:

Exercise 14.3

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In the next section, we will see implications.

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Chapter 14.1 - Negation of A Statement

In the previous section, we saw what a "statement" is. In this section, we will see new statements from old.

Some basic points can be written in 3 steps:
1. We are discussing about mathematical reasoning.
• We saw “statements” which are the basic units of mathematical reasoning.
• We want to know how statements help us in the process of mathematical reasoning.
2. The English mathematician George Boole put forward an effective technique to analyze statements.
• In this technique, we ask two questions:
   ♦ What happens when the statement is true ?
   ♦ What happens when the statement is false ?
3. Finding answers to the above two questions is a major step in mathematical reasoning.

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So our next aim is to learn about Negation of a statement. This can be written in 3 steps:
1. The denial of a statement is called the negation of the statement.
2. Let us see an example. It can be written in 3 steps:
(i) Consider the following statement:
p: New Delhi is a city.
(ii) The negation of this statement can be written in any one of the three ways shown below:
• It is not the case that New Delhi is a city
• It is false that New Delhi is a city
• New Delhi is not a city.
(iii) If p is a statement, then negation of p is also a statement.
   ♦ It is denoted as ~p.
   ♦ It is read as ‘not p’.
3. ~p helps us to analyze p in a better way. This can be demonstrated using some examples.
Example 1:
This can be written in 4 steps:
(i) Consider the statement:
p: Every one in Germany speaks German.   
(ii) We want ~p.
• p tells us that all people living in Germany speak German.
• ~p must tell us that all people living in Germany does not speak German.
(iii) This ~p can be achieved by saying that there is at least one person in Germany, who does not speak that language.
• So we get:
~p: There is at least one person in Germany who does not speak German.
(iv) p and ~p enable us to think of various possibilities.

Example 2:
This can be written in 4 steps:
(i) Consider the statement:
p: Both the diagonals of a rectangle have the same length.   
(ii) We want ~p.
• p tells us that: Take any rectangle. Both it's diagonals will be of the same length.
• ~p must tell us that the statement is not true for any rectangle.
(iii) This ~p can be achieved by saying that there is at least one rectangle where both diagonals are not the same.
• So we get:
~p: There is at least one rectangle where both diagonals are not of the same length.
(iv) p and ~p enable us to think of various possibilities.

Example 3:
This can be written in 4 steps:
(i) Consider the statement:
p: √7 is rational.   
(ii) We want ~p.
• p tells us that: √7 is rational.
• ~p must tell us that the statement is not true.
(iii) This ~p can be achieved by saying that it is not rational.
• So we get:
~p: √7 is not rational.
(iv) p and ~p enable us to think of various possibilities.

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Now we will see some solved examples.

Solved example 14.2
Write the negation of the following statements and check whether the resulting statements are true.
(i) Australia is a continent.
(ii) There does not exist a quadrilateral which has all it’s sides equal.
(iii) The sum of 3 and 4 is 9.
Solution:
Part (i):
1. Consider the statement:
p: Australia is a continent.
2. The negation can be written as:
~p: It is false that Australia is a continent.
OR
~p: Australia is not a continent.
3. ~p is false.

Part (ii):
1. Consider the statement:
p: There does not exist a quadrilateral which has all it’s sides equal.
2. We want ~p.
• p tells us that: Not even a single quadrilateral (with the given property) exists.
• ~p must tell us that "not even a single" is false.
3. This ~p can be achieved by saying that there is at least one quadrilateral.
• So we get:
~p: There is at least one quadrilateral which has all it’s sides equal.
4. ~p is true. (square is a quadrilateral which has all it’s sides equal)

Part (iii):
1. Consider the statement:
p: The sum of 3 and 4 is 9.
2. The negation can be written as:
~p: It is false that the sum of 3 and 4 is 9.
OR
~p: sum of 3 and 4 is not 9.
3. ~p is true.

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Compound statements

This can be explained in 4 steps:
1. Some mathematical statements are obtained by combining two or more simple statements.
• The simple statements are combined by using connecting words like “and”, “or” etc.,
2. This can be demonstrated using an example. It can be written in 2 steps:
(i) Consider the following statement:
p: There is something wrong with the bulb or with the wiring.
• This statement tells us that:
   ♦ The bulb is not turning on because,
   ♦ there is something wrong with the bulb.
   ♦ or there is something wrong with the wiring.
(ii) So the given statement is in fact formed by two statements:
q: There is something wrong with the bulb.
r: There is something wrong with the wiring.
   ♦ q and r are connected by “or”.
3. Another example:
(i) Consider the two statements:
p: 7 is an odd number.
q: 7 is a prime number.
• These two simple statements can be connected by the word “and”.
• We get:
r: 7 is both an odd number and a prime number.
4. A compound statement is a statement which is made up of two or more simple statements. Each of the simple statements is called a component statement.

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Let us see some solved examples:
Solved example 14.3
Find the component statements of the following compound statements.
(i) The sky is blue and the grass is green.
(ii) It is raining and it is cold.
(iii) All rational numbers are real and all real numbers are complex.
(iv) 0 is a positive number or a negative number.
Solution:
Part (i):
1. We have:
p: The sky is blue and the grass is green.
2. The component statements are:
q: The sky is blue.
r: The grass is green.
   ♦ q and r are connected by “and”.  

Part (ii):
1. We have:
p: It is raining and it is cold.
2. The component statements are:
q: It is raining.
r: It is cold.
   ♦ q and r are connected by “and”.  

Part (iii):
1. We have:
p: All rational numbers are real and all real numbers are complex.
2. The component statements are:
q: All rational numbers are real.
r:  All real numbers are complex.
   ♦ q and r are connected by “and”.

Part (iv):
1. We have:
p: 0 is a positive number or a negative number.
2. The component statements are:
q: 0 is a positive number.
r: 0 is a negative number.
   ♦ q and r are connected by “or”.

Solved example 14.4
Find the component statements of the following compound statements. Check each of the component statements and find whether they are true or false.
(i) A square is a quadrilateral and it’s four sides equal.
(ii) All prime numbers are either even or odd
(iii) A person who has taken Mathematics or Computer science can go for MCA
(iv) Chandigarh is the capital of Haryana and UP.
(v) 2 is a rational number or an irrational number.
(vi) 24 is a multiple of 2, 4 and 8.
Solution:
Part (i):
1. We have:
p: A square is a quadrilateral and it’s four sides equal.
2. The component statements are:
q: A square is a quadrilateral.
r: A square has all four sides equal.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is true.
• r is also true.

Part (ii):
1. We have:
p: All prime numbers are either even or odd
2. The component statements are:
q: All prime numbers are even numbers.
r: All prime numbers are odd numbers.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is false.
• r is false.

Part (iii):
1. We have:
p: A person who has taken Mathematics or Computer science can go for MCA.
2. The component statements are:
q: A person who has taken Mathematics can go for MCA.
r: A person who has taken Computer science can go for MCA.
   ♦ q and r are connected by “or”.
3. Checking component statements:
• q is true.
• r is true.

Part (iv):
1. We have:
p: Chandigarh is the capital of Haryana and UP.
2. The component statements are:
q: Chandigarh is the capital of Haryana.
r: Chandigarh is the capital of UP.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is true.
• r is false.

Part (v):
1. We have:
p: √2 is a rational number or an irrational number.
2. The component statements are:
q: √2 is a rational number.
r: √2 is an irrational number.
   ♦ q and r are connected by “or”.
3. Checking component statements:
• q is false.
• r is true.

Part (vi):
1. We have:
p: 24 is a multiple of 2, 4 and 8.
2. The component statements are:
q: 24 is a multiple of 2.
r: 24 is a multiple of 4.
s: 24 is a multiple of 8.
   ♦ q, r and s are connected by “and”.
3. Checking component statements:
q is true.
• r is true.
• s is true

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• We see that:
Component statements are connected by words like “and”, “or” etc., These words have special meanings in mathematics. We will see it in the next section.

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The link to a few more solved examples is given below:

Exercise 14.2

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In the next section, we will see special words/phrases.

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