Question

1) Form the biconditional statement $p\leftrightarrow q,$ where

i) $p:$ The unit digit of an integer is zero.
$q:$ It is divisible by 5.

2) Form the biconditional statement $p\leftrightarrow q,$ where

ii) $p:$ A natural number $n$ is odd.

$q:$ Natural number $n$ is not divisible by 2.

3) Form the biconditional statement $p\leftrightarrow q,$ where

iii) $p:$ A triangle is an equilateral triangle.

$q:$ All three sides of a triangle are equal.

Answer 1

Given

$p:$ The unit digit of an integer is zero.

$q:$ It is divisible by 5.

$\because$ If $p$ and $q$ are any two statements, then the compound statement ${}^{\prime \prime }p\text{if and only if q''}$ formed by joining $p$ and $q$ by a connective ''if and only if '' is called biconditional statement.

$\therefore$ The biconditional $p\leftrightarrow q$ of the given statements is

The unit digit of an integer is zero, if and only if it is divisible by 5.

2) Given

$p:$ A natural number $n$ is odd.

$q:$ Natural number $n$ is not divisible by 2.

$\because$ If $p$ and $q$ are any two statements, then the compound statement ${}^{\prime \prime }p\text{if and only if q''}$ formed by joining $p$ and $q$ by a connective ''if and only if '' is called biconditional statement.

$\therefore$ The biconditional $p\leftrightarrow q$ of the given statements is

A natural number $n$ is odd, if and only if it is not divisible by 2.

3) Given

iii) $p:$ A triangle is an equilateral triangle.

$q:$ All three sides of a triangle are equal.

$\because$ If $p$ and $q$ are any two statements, then the compound statement ${}^{\prime \prime }p\text{if and only if q''}$ formed by joining $p$ and $q$ by a connective ''if and only if '' is called biconditional statement.

$\therefore$ The biconditional $p\leftrightarrow q$ of the given statements is

A triangle is an equilateral triangle, if and only if all three sides of a triangle are equal.