Mock Test 2 — Mathematical Reasoning

The statement $(\sim(p \Leftrightarrow q)) \Leftrightarrow ((p \wedge \sim q) \vee (\sim p \wedge q))$ is:

The Boolean expression $\sim(p \vee q) \vee (\sim p \wedge q)$ is equivalent to:

Let $p$: "A quadrilateral is a square", $q$: "Its diagonals are equal", $r$: "Its diagonals are perpendicular". The statement $p \Rightarrow (q \wedge r)$ is:

The statement $(p \Rightarrow q) \Leftrightarrow (\sim q \Rightarrow \sim p)$ is:

The negation of the compound statement "$\sqrt{2}$ is irrational and 2 is prime" is:

Consider the argument: "If a number is divisible by 6, then it is divisible by both 2 and 3. The number $n$ is divisible by 2 and 3. Therefore $n$ is divisible by 6." This argument is:

Which of the following is a valid argument form?

The Boolean expression $((p \cdot q')' \cdot (p' + q))'$ simplifies to:

If $p$ is the statement "Ravi is hardworking" and $q$ is "Ravi will get a job", then the statement "Ravi will get a job if and only if he is hardworking" is:

The contrapositive of "$(p \Rightarrow q) \Rightarrow r$" is:

Consider 4 atomic statements $p, q, r, s$. How many rows are there in the corresponding truth table?

The number of distinct truth tables (functions from $\{T, F\}^2 \to \{T, F\}$) that can be defined for two atomic statements $p, q$ is $2^k$. Find $k$.

How many of the following are tautologies?
(i) $p \vee \sim p$, (ii) $p \wedge \sim p$, (iii) $(p \Rightarrow q) \vee (q \Rightarrow p)$, (iv) $(p \Rightarrow q) \Leftrightarrow (\sim p \vee q)$

Assertion (A): The statement "If $n$ is an integer and $n^2$ is even, then $n$ is even" can be efficiently proven by contrapositive.
Reason (R): The contrapositive "If $n$ is odd, then $n^2$ is odd" is straightforward to prove directly using $n = 2k+1$.

Assertion (A): For any propositions $p$ and $q$, the proposition $p \Rightarrow q$ is logically equivalent to $\sim p \vee q$.
Reason (R): Truth table analysis confirms that $p \Rightarrow q$ and $\sim p \vee q$ have identical truth values for all combinations of $p$ and $q$.