# Questions & Answers of Mathematical Logic

#### Topics of Mathematical Logic 25 Question(s)

Question No. 1

The statement $\style{font-family:'Times New Roman'}{\left(\neg p\right)\Rightarrow\left(\neg q\right)}$ is logically equivalent to which of the statments below?

I. $\style{font-family:'Times New Roman'}{p\Rightarrow q}$

II. $\style{font-family:'Times New Roman'}{q\Rightarrow p}$

III. $\style{font-family:'Times New Roman'}{\left(\neg q\right)\vee p}$

IV. $\style{font-family:'Times New Roman'}{\left(\neg p\right)\vee q}$

Question No. 2

Consider the first-order logic sentence $\style{font-family:'Times New Roman'}{F:\forall x\left(\exists yR\left(x,y\right)\right)}$. Assuming non-empty logical domains, which of the sentence below are implied by F?

$\style{font-family:'Times New Roman'}{\mathrm I.\;\exists y\left(\exists xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{II}.\;\exists y\left(\forall xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{III}.\;\forall y\left(\exists xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{IV}.\;\neg\exists y\left(\forall y\neg R\left(x,y\right)\right)}$

Question No. 29

Let p,q, and r be proposition and the expression $\style{font-family:'Times New Roman'}{\left(p\rightarrow q\right)\rightarrow r}$ be a contradiction. Then, the expression $\style{font-family:'Times New Roman'}{\left(r\rightarrow p\right)\rightarrow q}$ is

Question No. 111

Let p, q, r, denote the statements "It is raining". "It is cold", and "It is pleasant", respectivelyThen the statement "It is not raining and it is pleasant, and if it is raining only if it is raining and it is cold" is represented by

Question No. 111

Consider the following expressions:

(i) false

(ii) Q

(iii) true

(iv) PQ

(v) ¬QP

The number of expressions given above that are logically implied by P(P Q) is__________ .

Question No. 19

Which one of the following is NOT equivalent to $p↔q$?

Question No. 111

Consider the following two statements.
S1: If a candidate is known to be corrupt, then he will not be elected
S2: If a candidate is kind, he will be elected
Which one of the following statements follows from S1 and S2 as per sound inference rules of logic?

Question No. 11

Consider the statement
“Not all that glitters is gold”
Predicate glitter(x) is true if x glitters and predicate gold(x) is true if x is gold. Which one of the following logical formulae represents the above statement?

Question No. 63

Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?

Question No. 163

Which one of the following Boolean expressions is NOT a tautology?

Question No. 211

Consider the following statements:
P: Good mobile phones are not cheap
Q: Cheap mobile phones are not good
L: P implies Q
M: Q implies P
N: P is equivalent to Q
Which one of the following about L, M, and N is CORRECT?

Question No. 263

The CORRECT formula for the sentence, “not all rainy days are cold” is

Question No. 27

What is the logical translation of the following statement?
“None of my friends are perfect.”

Question No. 47

Which one of the following is NOT logically equivalent to $¬\exists x\left(\forall y\left(\alpha \right)\wedge \forall z\left(\beta \right)\right)$ ?

Question No. 1

Consider the following logical inferences.

I1: If it rains then the cricket match will not be played.
The cricket match was played.
Inference: There was no rain.
I2: If it rains then the cricket match will not be played.
It did not rain.
Inference: The cricket match was played.

Which of the following is TRUE?

Question No. 13

What is the correct translation of the following statement into mathematical logic?

“Some real numbers are rational”

Question No. 30

Which one of the following options is CORRECT given three positive integers x, y and z , and a predicate

$P\left(x\right)=¬\left(x=1\right)\wedge \forall y\left(\exists z\left(x={y}_{*}z\right)⇒\left(y=x\right)\vee \left(y=1\right)\right)$

Question No. 30

Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. which one of the statements below expresses best the meaning of the formula $\forall x\exists y\exists t\left(¬F\left(x,y,t\right)\right)$?

Question No. 23

Which one of the following is the most appropriate logical formula to represent the statement:

“Gold and silver ornaments are precious”

The following notations are used:

G(x): x is a gold ornament.
S(x): x is a silver ornament.
P(x): x is precious.

Question No. 24

The binary operation $\tiny\Box$ is defined as follows

 P Q P $\tiny\Box$ Q T T T T F T F T F F F T

Which one of the following is equivalent to P $\vee$ Q?

Question No. 26

Consider the following well-formed formulae:

I. $¬\forall x\left(P\left(x\right)\right)$
II. $¬\exists x\left(P\left(x\right)\right)$
III. $¬\exists x\left(¬P\left(x\right)\right)$
IV. $\exists x\left(¬P\left(x\right)\right)$

Which of the above are equivalent?

Question No. 30

Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton, and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent (a, b) means a and b are equivalent. Which of the following first order logic statements represents the following:
Each finite state automaton has an equivalent pushdown automaton.

Question No. 31

P and Q are two propositions. Which of the following logical expressions are equivalent?
I. P $\vee$ $~$ Q
II.$~$ ($~$ P $\wedge$ Q)
III. (P $\wedge$ Q)$\vee$ (P $\wedge$ $~$ Q) $\vee$ ($~$ P $\wedge$ $~$ Q)
IV. (P $\wedge$ Q) $\vee$ (P $\wedge$ $~$ Q) $\vee$ ($~$ P $\wedge$ Q)