Webpage for Mathematical Logic

Important Question Types:

1. Checking validity of First Order Logic Statements
2. Checking validity of Propositional Logic

# Recent questions tagged mathematical-logic

1
Which of the following pairs of propositions are not logically equivalent? $((p \rightarrow r) \wedge (q \rightarrow r))$ and $((p \vee q) \rightarrow r)$ $p \leftrightarrow q$ and $(\neg p \leftrightarrow \neg q)$ $((p \wedge q) \vee (\neg p \wedge \neg q))$ and $p \leftrightarrow q$ $((p \wedge q) \rightarrow r)$ and $((p \rightarrow r) \wedge (q \rightarrow r))$
2
If $f(x)=x$ is my friend, and $p(x) =x$ is perfect, then correct logical translation of the statement “some of my friends are not perfect” is ______ $\forall _x (f(x) \wedge \neg p(x))$ $\exists _x (f(x) \wedge \neg p(x))$ $\neg (f(x) \wedge \neg p(x))$ $\exists _x (\neg f(x) \wedge \neg p(x))$
3
What kind of clauses are available in conjunctive normal form? Disjunction of literals Disjunction of variables Conjunction of literals Conjunction of variables
4
Which of the following is FALSE? $Read\ \wedge as\ AND, \vee\ as\ OR, \sim as\ NOT, \rightarrow$ as one way implication and $\leftrightarrow$ as two way implication? $((x\rightarrow y)\wedge x)\rightarrow y$ $((\sim x\rightarrow y)\wedge (\sim x\wedge \sim y))\rightarrow x$ $(x\rightarrow (x\vee y))$ $((x\vee y)\leftrightarrow (\sim x\vee \sim y))$
5
In propositional logic, which of the following is equivalent to $p \rightarrow q$? $\sim p\rightarrow q$ $\sim p \vee q$ $\sim p \vee \sim q$ $p\rightarrow \sim q$
1 vote
6
Which of the following statements is false? $(P\land Q)\lor(\sim P\land Q)\lor(P \land \sim Q)$ is equal to $\sim Q\land \sim P$ $(P\land Q)\lor(\sim P\land Q)\lor(P \wedge \sim Q)$ is equal to $Q\lor P$ $(P\wedge Q)\lor (\sim P\land Q)\lor(P \wedge \sim Q)$ is equal to $Q\lor (P\wedge \sim Q)$ $(P\land Q)\lor(\sim P\land Q)\lor (P \land \sim Q)$ is equal to $P\lor (Q\land \sim P)$
7
Which of the following propositions is tautology? $(p\lor q)\to q$ $p\lor (q\to p)$ $p\lor (p\to q)$ Both (B) and (C)
8
Which one is the correct translation of the following statement into mathematical logic? “None of my friends are perfect.” $\neg\:\exists\:x(p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land q(x))$ $\exists\:x(\neg\:p(x)\land\neg\:q(x))$ $\exists\:x(p(x)\land\neg\:q(x))$
9
Which of the following statements is true? The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2} \wedge \dots \wedge S_{n}\rightarrow S$ is satisfiable. The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if ... logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2}\wedge \dots \wedge S_{n}\wedge S$ is inconsistent.
10
Match the following : ...
11
Given that $B(a)$ means “$a$ is a bear” $F(a)$ means “$a$ is a fish” and $E(a,b)$ means “$a$ eats $b$” Then what is the best meaning of $\forall x [F(x) \to \forall y(E(y,x)\rightarrow b(y))]$ Every fish is eaten by some bear Bears eat only fish Every bear eats fish Only bears eat fish
1 vote
12
In the land of Twitter, there are two kinds of people: knights (also called outragers), who always tell the truth, and knaves (also called trolls), who always lie. It so happened that a person with handle @anand tweeted something offensive. It was not known ... Suspect $3:$ My lawyer always tells the truth. Which of the above suspects are innocent, and which are guilty? Explain your reasoning.
13
“Not every satisfiable logic is valid” Representation of it will be $1)\sim \left ( \forall S(x)\rightarrow V(x) \right )$ or $2)\sim \left ( \forall S(x)\vee V(x) \right )$ Among $1)$ and $2)$, which one is correct? and why?
14
Read the statements: All women are entrepreneurs. Some women are doctors. Which of the following conclusions can be logically inferred from the above statements? All women are doctors All doctors are entrepreneurs All entrepreneurs are women Some entrepreneurs are doctors Why here $2)$ ... ans?? Is it because , if we make set of doctor as 0, then All doctors are entrepreneurs is meaningless.
1 vote
15
The notation $\exists ! x P(x)$ denotes the proposition “there exists a unique $x$ such that $P(x)$ is true”. Give the truth values of the following statements : I)${\color{Red} {\exists ! x P(x)}} \rightarrow \exists x P(x)$ II)${\color{Red} {\exists ! x\sim P(x)}} \rightarrow \neg \forall x P(x)$ What will be answer here?? Is the assumption only for left hand side and not right hand side??
16
Are these propositions? 1.This sentence is true 2.This sentence is false Aren’t these liar paradox?
1 vote
17
Represent these two statement in first order logic: $A)$ Only Alligators eat humans $B)$ Every Alligator eats humans Is Every represents $\equiv \exists$ and Only represents $\equiv \forall$ ?? Can we differentiate it with verb ‘eat’ and ‘eats’??
18
Consider the following first order logic statement $I)\forall x\forall yP\left ( x,y \right )$ $II)\forall x\exists yP\left ( x,y \right )$ $III)\exists x\exists yP\left ( x,y \right )$ $III)\exists x\forall yP\left ( x,y \right )$ ... $II)$ is true , then $III),IV)$ is true $B)$ If $IV)$ is true , then $II),III)$ is true $C)$ None of these
19
Determine whether each of these statements is true or false. $0$ $\epsilon$ $\phi$ $\phi$ $\epsilon$ {$0$} {$0$} $\subset$ {$\phi$} $\phi$ $\subset$ {$0$} {$0$} $\epsilon$ {$0$} {$0$} $\subset$ {$0$} {$\phi$} $\subseteq$ {$\phi$}
20
Prove that these four statements about the integer $n$ are equivalent: $n^2$is odd, $1−n$ is even, $n^3$ is odd, $n^2+1$ is even.
21
Prove that if $n$ is an integer, these four statements are equivalent: $n$ is even, $n+1$ is odd, $3n+1$isodd, $3n$ is even.
Prove that at least one of the real numbers $a_1,a_2,...,a_n$ is greater than or equal to the average of these numbers.What kind of proof did you use?
Show that the propositions $p1,p2,p3,p4,$ and $p5$ can be shown to be equivalent by proving that the conditional statements $p1 \rightarrow p4$ , $p3 \rightarrow p1$ ,$p4 \rightarrow p2$ ,$p2 \rightarrow p5$, and $p5 \rightarrow p3$ are true.