Tag: logical equivalence

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Questions Related to logical equivalence

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following proposition is a contradiction?

  1. $(\sim p\vee \sim q)\vee (p\vee \sim q)$
  2. $(p\rightarrow q)\vee (p\wedge \sim q)$
  3. $(\sim p\wedge q)\wedge (\sim q)$
  4. $(\sim p\wedge q)\vee (\sim q)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Contradiction is a preposition which is always false(F).
Here $\sim \equiv negation$ and $\wedge \equiv AND$
Let $ x=(\sim p \wedge q) \wedge (\sim q)$
If $p=F$ and $q=F$ then $x=F$
If $p=F$ and $q=T$ then $x=F$
If $p=T$ and $q=F$ then $x=F$
If $p=T$ and $q=T$ then $x=F$
Hence option (c) is correct

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p,q,r$ are $3$ statement such that $(p\rightarrow q)\wedge (q\rightarrow r)\Rightarrow (p\rightarrow r)$ is 

  1. Tautology

  2. Contradiction
  3. $P\wedge q$
  4. $p\wedge (\sim q)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression (p -> q) ^ (q -> r) -> (p -> r) is the Law of Hypothetical Syllogism, which is a fundamental tautology in propositional logic.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p,q,r$ are $3$ statement such that $(p \rightarrow q)\wedge (q \rightarrow r)\Rightarrow (P \rightarrow r)$ is

  1. Tautology

  2. Contradiction
  3. $P \wedge q$
  4. $p \wedge (\sim q)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is identical to the previous question; the expression (p -> q) ^ (q -> r) -> (p -> r) is a tautology.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

Which of the following is a tautology?

  1. $p\wedge (\sim p)$
  2. $p\wedge c$
  3. $p\vee t$
  4. $p\wedge p$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A tautology is a compound statement that is always true regardless of the truth values of its components. For option C, the disjunction of any proposition p with a true statement t always results in true, satisfying the definition of a tautology.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

The only statement among the followings that is a tautology is

  1. $A\vee(A\wedge B)$
  2. $[A\wedge (A\rightarrow B)]\rightarrow B$
  3. $B\rightarrow [A\wedge (A\rightarrow B)]$
  4. $A\wedge (A\vee B)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Option B represents modus ponens, which is a universally valid argument form and thus a tautology. By constructing a truth table, we find that the conditional statement always evaluates to true.

Multiple choice logical equivalence mathematical logic discrete mathematics business maths maths

$p,q,r$ are $3$ statements such that $\left(p\rightarrow q\right)\wedge \left(q\rightarrow r\right)=Rightarrow \left(p\rightarrow r\right)$ is

  1. $Tautology$
  2. $Contradiction$
  3. $P\wedge q$
  4. $p\wede(\sim q)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression (p -> q) ^ (q -> r) -> (p -> r) is a classic logical law known as the Hypothetical Syllogism. A truth table shows that it evaluates to True for all possible truth values of p, q, and r, making it a tautology.