Recent questions tagged mathematical-logic

Given $(P \to (Q \land R)) \land (Q \to R)$, what is the conjunctive normal form?$(\neg P \lor Q)\land(\neg P \lor R)\land(\neg Q \lor R)$ $(P \lor Q)\land(P \lor R)\land...

Which of the following statements are true if universe of discourse for all variables consist of all integers?$\forall n \exists m\left(n^{2} < m \right)$$\exists n \exis...

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason RAssertion A: If $L$ is regular, then its compliment $L^{\prime}$ is ne...

Let C(x,y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being g...

For the statement $$P \rightarrow Q$$Which of the following statements below is/are correct?If the hypothesis is true and the conclusion is false, then the conditional st...

For two different persons $x$ and $y$, the predicate $M(x, y)$ denotes that $x$ knows $y$. Consider the following statement.There is a person who does not know anyone els...

Let $P(x)$ be a predicate.Which of the following statements is/are NOT valid in first-order logic?$\forall x\; P(x) \Rightarrow \exists x\; P(x)$ $\exists x\; P(x) \Right...

Which of the following statements is NOT true?(The names of the predicates are intuitive.)$\forall x\ \forall y\ \emph{Classmate(x,y)} \Rightarrow \emph{Classmate(y,x)}$ ...

Assume that a Creative $(C)$ person will Succeed $(S)$ if the person is also Disciplined $(D)$, but will not succeed otherwise. Now, consider the following statements:$C ...

Sentence $X$ is said to entail Sentence $Y$ if whenever $X$ is TRUE, $Y$ also must hold TRUE.Which of the following statements is/are correct if $X$ entails $Y$?$X \Right...

Consider the following Boolean expression of a function $F$ :\[F(P, Q)=(\bar{P}+Q) \oplus(\bar{P} Q)\]Which of the following expressions is/are equivalent to $F$ ?$\overl...

For two different persons $x$ and $y$, the predicate $M(x,y)$ denotes that $x$ knows $y$.Consider the following statement.$\emph{"There is a person who does not know anyo...

"When it is raining, peacocks dance.''Based only on this sentence, which one of the following options is necessarily true?Peacocks dance only when it is raining. When pea...

'When the teacher is in the room, all students stand silently.'If the above statement is true, which one of the following statements is not necessarily true?If any studen...

Consider the formula:∀x(P(x)→Q(x)) ∧ ∀x(Q(x)→R(x)) → ∀x(P(x)→R(x))Which is true?A. TautologyB. ContingentC. ContradictionD. Unsatisfiable

Consider the $\operatorname{ring} \mathcal{C}(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$, with pointwise addition and multiplication. For $A \subset...

Prove or disprove each of the statements below.Let $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}$ be a continuous function that takes both positive and negative values. T...

Consider the following if $p$ and $q$ are two statements.$\sim(\mathrm{p} \wedge \mathrm{q}) \equiv \sim \mathrm{p} \vee \sim \mathrm{q}$$\sim(\mathrm{p} \vee \mathrm{q})...

If the universe of disclosure is set of integers, then which of the followings are TRUE ?$\forall \mathrm{n} \exists \mathrm{m} \: \left(\mathrm{n}^{2}<\mathrm{m}\right)$...

Given below are two statements:$\text{Statement (I)}$: Truth and falsehood are attributes of individual propositions.$\text{Statement (II)}$: Validity can be attributed t...