Let $\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$ be such that $(p \wedge q) \Delta ((p \vee q) \Rightarrow q)$ is a tautology. Then $\Delta$ is equal to

The negation of the statement "If a number is divisible by $15$,then it is divisible by $5$ and $3$" is:

Which of the following statements is/are not true?
$I$) If $1$ is not a prime number, then $2$ is not a prime number.
$II$) $e$ is a vowel and $12 \times 3 = 36$.
$III$) It is not true that $14$ is a composite number and $3$ is an even number.
$IV$) $\sqrt{5}$ is an irrational number, but $3 + \sqrt{5}$ is a complex number.

Write the component statements of the following compound statement and check whether the compound statement is true or false.
$0$ is less than every positive integer and every negative integer.

Let $p, q, r$ be three statements such that the truth value of $(p \wedge q) \rightarrow (\sim q \vee r)$ is $F$. Then the truth values of $p, q, r$ are respectively:

If $p: \text{A man is happy}$,$q: \text{A man is rich}$,then the symbolic form of '$\text{A man is neither happy nor rich}$' is