Mathematics MATHEMATICAL REASONING - Implications and Contrapositive and converse
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### Topics Covered

star Implications
star Contrapositive and converse

### Implications

color{green} ✍️ In this Section, we shall discuss the implications of color(blue)"“if-then”, “only if”" and color(blue)"“if and only if ”." The statements with color(blue)"“if-then”" are very common in mathematics.

=>color(red)("For example, consider the statement.")

 \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" r: If you are born in some country, then you are a citizen of that country."

When we look at this statement, we observe that it corresponds to two statements p and q given by

 \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" p : you are born in some country."

 \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q : you are citizen of that country."

Then the sentence color(blue)"“if p then q”" says that in the event if p is true, then q must be true.

One of the most important facts about the sentence "“if p then q”" is that it does not say any thing (or places no demand) on q when p is false.

For example, if you are not born in the country, then you cannot say anything about q. To put it in other words” not happening of p has no effect on happening of q.

Another point to be noted for the statement "“if p then q”" is that the statement does not imply that p happens.

There are several ways of understanding “if p then q” statements.

=> We shall illustrate these ways in the context of the following statement.

 \ \ \ \ \ \ \ \ \ \ \ \ \color{green} " r: If a number is a multiple of 9, then it is a multiple of 3."
Let p and q denote the statements

 \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" p : a number is a multiple of 9."

 \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q: a number is a multiple of 3."

Then, if p then q is the same as the following:
color(red)(1.) \ \p \ \ "implies" \ \ q is denoted by p ⇒ q. The symbol ⇒ stands for implies. This says that a number is a multiple of 9 implies that it is a multiple of 3.

color(red)(2.) \ \ p is a sufficient condition for q.

This says that knowing that a number as a multiple of 9 is sufficient to conclude that it is a multiple of 3.

color(red)(3.) \ \p only if q.
This says that a number is a multiple of 9 only if it is a multiple of 3.

color(red)(4.) \ \q is a necessary condition for p.

This says that when a number is a multiple of 9, it is necessarily a multiple of 3.

color(red)(5.) \ \∼q " implies" ∼p.

This says that if a number is not a multiple of 3, then it is not a multiple of 9.

### Contrapositive and converse

Contrapositive and converse are certain other statements which can be formed from a given statement with color(blue)"“if-then”."

=>color(red)("For example,") let us consider the following "“if-then”" statement.

color(green)("If the physical environment changes, then the biological environment changes.")

Then the contrapositive of this statement is

color(orange)"If the biological environment does not change, then the physical environment does not change."

color(red)"Note" that both these statements convey the same meaning.

color(blue)"“Converse”."

The color{blue} ul(mathtt(Converse)) of a given statement “if p, then q” is if q, then p.
Q 3101701628

Write the contracepostive of the following statement:
(i) If a number is divisible by 9, then it is divisible by 3.
(ii) If you are born in India, then you are a citizen of India.
(iii) If a triangle is equilateral, it is isosceles.

Solution:

The contrapositive of the these statements are
(i) If a number is not divisible by 3, it is not divisible by 9.
(ii) If you are not a citizen of India, then you were not born in India.
(iii) If a triangle is not isosceles, then it is not equilateral.
The above examples show the contrapositive of the statement if p, then q is “if ∼q,
then ∼p”.
Next, we shall consider another term called converse.
The converse of a given statement “if p, then q” is if q, then p.

For example, the converse of the statement
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p: If a number is divisible by 10, it is divisible by 5 is"
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q: If a number is divisible by 5, then it is divisible by 10"
Q 3111701629

Write the converse of the following statements.
(i) If a number n is even, then n^2 is even.
(ii) If you do all the exercises in the book, you get an A grade in the class.
(iii) If two integers a and b are such that a > b, then a – b is always a positive integer.

Solution:

The converse of these statements are
(i) If a number n^2 is even, then n is even.
(ii) If you get an A grade in the class, then you have done all the exercises of the book.
(iii) If two integers a and b are such that a – b is always a positive integer, then a > b.
Let us consider some more examples.
Q 3111801720

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:

(i) The component statements are given by
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p : Triangle ABC is equilateral."
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q : Triangle ABC is Isosceles."
Since an equilateral triangle is isosceles, we infer that the given compound statement
is true.
(ii) The component statements are given by
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p : a and b are integers."
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q : ab is a rational number."
since the product of two integers is an integer and therefore a rational number, the
compound statement is true.
‘If and only if’, represented by the symbol ‘⇔‘ means the following equivalent forms
for the given statements p and q.
(i) p if and only if q
(ii) q if and only if p
(iii) p is necessary and sufficient condition for q and vice-versa
(iv) p ⇔ q
Consider an example.
Q 3141801723

Given below are two pairs of statements. Combine these two statements using “if and only if ”.
(i) p: If a rectangle is a square, then all its four sides are equal.
q: If all the four sides of a rectangle are equal, then the 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 its digits is divisible by 3.

Solution:

(i) A rectangle is a square if and only if all its four sides are equal.
(ii) A number is divisible by 3 if and only if the sum of its digits is divisible by 3.