Question

If $p$ and $q$ are simple propositions, then $p\iff ~q$ is true when

• A. $p$is true and $q$is true
• B. Both $p$ and $q$ are false
• C. $p$is false and $q$is true
• D. None of these

Answer 1

The correct option is C

$p$is false and $q$is true

Explantion for the correct option:

Step-1: Definition of Logical negation and Bi-conditional:

Logical Negation: Logical negation is defined as when we apply negation to a proposition then it gives the opposite value as output for the input. If $p$ is True then $~p$is False and vice versa.

Notation: Negation is denoted by the symbol $~$

Bi-conditional:

Let, $p$ and $q$are two simple propositions. If $p$ and $q$ both the propositions are either True or False then Bi-conditional gives the output as True otherwise if anyone is False then the output is False.

Notation: Bi-conditional is denoted by the symbol $\iff$

Step-2: Truth Table:

The truth table is used to perform logical operations on given propositions. It is used to check whether the given propositional statement is either True or False. Using some of the logical operations like AND, OR, NOR, Conditional, and Bi-conditional.

Given $p$ and $q$ are both simple propositions

The truth table for $p\iff ~q$ is given below

$p$	$q$	$~q$	$p\iff ~q$
T	T	F	F
T	F	T	T
F	T	F	T
F	F	T	F

Where T is True and F is False.

From the above table, we can conclude $p\iff ~q$is True if $p$is False and $q$ is True.

Hence option 'C' is the correct answer.