List of top Mathematics Questions on Mathematical Logic

Consider the following logical statements:
R: If \(p \rightarrow q\) is false, then \(p \vee q\) is false.
S: If \(p \leftrightarrow q\) is false, then \(p \vee q\) is false.
Evaluate the truth values of R and S.

If the statement $(p \wedge q) \rightarrow (r \vee \neg s)$ is False (F), then the truth values of $p, q, r$ and $s$ are respectively

Identify the truth value of the statement pattern \( (p \land \sim q) \rightarrow r \).

\( \int_{0}^{1} \log (\frac{1}{x} - 1) dx = \)}

The differential equation whose solution is \( Ax^2 + By^2 = 1 \), where A and B are arbitrary constants is of}

$\int \frac{\sin 2x}{(a+b\cos x)^2} dx =$

If $3 \sin \alpha = 5 \sin \beta$, then $\tan \left( \frac{\alpha+\beta}{2} \right) + \tan \left( \frac{\alpha-\beta}{2} \right) =$

The circumradius of the triangle formed by the lines $xy + 2x + 2y + 4 = 0$ and $x + y + 2 = 0$ is

If $\bar{p} = 2\hat{\text{i}} + \hat{\text{k}}, \bar{q} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}}, \bar{r} = 4\hat{\text{i}} - 3\hat{\text{j}} + 7\hat{\text{k}}$ and a vector $\bar{\text{m}}$ is such that $\bar{\text{m}} \times \bar{q} = \bar{r} \times \bar{q}, \bar{\text{m}} \cdot \bar{p} = 0$, then $\bar{\text{m}} = .......$

The number of ordered triplets of the truth values of \( p, q, r \) and such that the truth value of the statement \[ (p \lor q) \land (p \lor r) \implies (q \lor r) \text{ is True, is equal to:} \]

If \( p \to (\sim p \lor q) \) is false, then the truth values of \( p \) and \( q \) are, respectively

The logical statement \((p \to q) \land (q \to \sim p)\) is equivalent to

Negation of $(p \land q) \rightarrow (\sim p \lor r)$ is

p : It rains today
q : I am going to school
r : I will meet my friend
s : I will go to watch a movie.
Then symbolic form of the statement "If it does not rain today or I won't go to school, then I will meet my friend and I will go to watch a movie" is

The logical expression $p \wedge(\sim p \vee \sim q) \equiv$

The negation of a statement '$x \in A \cap B \rightarrow (x \in A \text{ and } x \in B)$' is

If $p, q$ are true statements and $r$ is false statement, then which of the following is correct.

Negation of the statement $\forall x \in R, x^2+1=0$ is

The statement $P(n) = 9^{n} - 8^{n}$, when divided by 8, always leaves the remainder

The negation of the compound proposition $p \vee (\neg p \vee q)$ is

If $P(n): 1 + 3 + 5 + \cdots + (2n-1) = n^2$ is

If $P(n)$ is a statement such that $P(3)$ is true. Assuming $P(k)$ is true $\Rightarrow P(k+1)$ is true for all $k \ge 3$, then $P(n)$ is true

$(p \land q) \lor (\sim p \land \sim q)$ is

Negation of "Paris is in France and London is in England" is

The equation of the circle circumscribing the triangle formed by the lines $x+y=6$, $2x+y=4$ and $x+2y=5$ is