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.
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.
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.
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.
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 rational number. An integer is a rational number.
• We can write:
If p happens, q is true. So r is true.
“If an 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
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.
The link to some solved examples is given below:
In the next section, we will see validating statements.
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