Question

Write the following in symbolic form:
A) Manisha does not live in Mumbai.
B) If a number ${n^2}$ is even, then $n$ is even.
C) Rohit is neither healthy nor wealthy.
D) If $\Delta ABC$ is right-angled at B, then, $A{B^2} + B{C^2} = A{C^2}$.
E) It is raining if and only if the weather is humid.

Answer 1

This is a problem of mathematical reasoning. We will start by writing each of these statements in their symbolic forms. For part (i), a negation sign will be used. For part (ii), an implicit sign will be used. For part (iii), the negation with and sign will be used, and so on.

Complete step-by-step answer:
(i)
Let us consider p = Manisha lives in Mumbai.
So, Manisha does not live in Mumbai is opposite of Manisha lives in Mumbai.
Thus, the symbolic form of the statement is
$\therefore \sim p$
Hence, the symbolic form of Manisha does not live in Mumbai is $\sim p$.
(ii)
Let us consider p = ${n^2}$ is even and q = n is even.
As we know that the square root of an even number is always even and the square root of an odd number is always odd.
Thus, the symbolic form of the statement is
$\therefore p \to q$
Hence, the symbolic form of if a number ${n^2}$ is even, then $n$ is even is $p \to q$.
(iii)
Let us consider p = Rohit is healthy and q = Rohit is wealthy.
So, the statement means Rohit is not healthy and not wealthy.
Thus, the symbolic form of the statement is
$\therefore \sim p \wedge \sim q$
Hence, the symbolic form of Rohit is neither healthy nor wealthy is $\sim p \wedge \sim q$.
(iv)
Let us consider p = $\Delta ABC$ is right-angled at B and q = $A{B^2} + B{C^2} = A{C^2}$.
As we know that the sum of squares of the adjacent side of the right-angle is equal to the square of the side opposite to the right angle.
Thus, the symbolic form of the statement is
$\therefore p \to q$
Hence, the symbolic form of if $\Delta ABC$ is right-angled at B, then, $A{B^2} + B{C^2} = A{C^2}$ is $p \to q$.
(iv)
Let us consider p = it is raining and q = the weather is humid.
The statement shows that if the weather is humid then it is raining.
It means it is true from both sides.
Thus, the symbolic form of the statement is
$\therefore p \leftrightarrow q$
Hence, the symbolic form of It is raining if and only if the weather is humid is $p \leftrightarrow q$.

Note: The important thing for students to do here is that they need to memorize the meanings of all the symbols because they will help in forming the truth tables for each of the statements. For example, in this problem, ~ stands for ‘not’, ∧ stands for ‘and’, ∨ stands for ‘or’, and so on.