a number is divisible by , it is divisible by is q: If a number is divisible by , then it is divisible by . Example Write the converse of the following statements. If a number n is even, then n is even. 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 If a number n is even, then n is even. 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. Example For each of the following compound statements, first identify the corresponding component statements. Then check whether the statements are true or not. If a triangle ABC is equilateral, then it is isosceles.
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.
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 .
p if and only if q q if and only if p MATHEMATICS (iii) p is necessary and sufficient condition for q and vice-versa (iv) p ⇔ q Consider an example. Example Given below are two pairs of statements. Combine these two statements using “if and only if ”. p: