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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