`star` Implications

`star` Contrapositive and converse

`star` Contrapositive and converse

`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.`

`=>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 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.

`=>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.

(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.

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.

(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.

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.

(i) If a triangle ABC is equilateral, then it is isosceles.

(ii) If a and b are integers, then ab is a rational number.

(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.

(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.

(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.