Identify the quantifier in the following statement and write the negation of the statement.
There exists a capital for every state in India.

The negation of the inverse of the statement $\sim p \lor \sim q$ is...

The inverse of the statement pattern $(p \vee q) \rightarrow (p \wedge q)$ is

Given below are two pairs of statements. Combine these two statements using "if and only if".
$p:$ 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 $3.$

The logical equivalent of $p \Leftrightarrow q$ is :-

Let $F_{1}(A, B, C) = (A \wedge \sim B) \vee [\sim C \wedge (A \vee B)] \vee \sim A$ and $F_{2}(A, B) = (A \vee B) \vee (B \rightarrow \sim A)$ be two logical expressions. Then ...... .