GATE CSE 1989 | Question: 3-v

Question

Which of the following well-formed formulas are equivalent?

• A. $P \rightarrow Q$
• B. $\neg Q \rightarrow \neg P$
• C. $\neg P \vee Q$
• D. $\neg Q \rightarrow P$

Comments

A,B and C are equivalent

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Note that, relating to conditional and biconditional statements, and their converse, inverse, or contrapositive: only a conditional and its contrapositive are equivalent (p → q ≡ ¬q → ¬p), and only a biconditional and its inverse are equivalent (p ↔ q ≡ ¬p ↔ ¬q).

Answer 1

• A. $P→Q   \equiv \neg P\vee Q$
• B. $¬Q→¬P\equiv Q\vee \neg P$
• C. $¬P\vee Q   \equiv \neg P \vee Q$

So, $A,B,C$ are equivalent .

Answer 2

• A. $P\rightarrow Q \equiv \neg P \vee Q$
• B. $\neg Q\rightarrow \neg P\equiv \neg(\neg Q)\vee \neg P \equiv \neg P \vee Q$
• C. $\neg P\vee Q$
• D. $\neg Q \rightarrow P\equiv \neg (\neg Q) \vee P \equiv P \vee Q$

So,  $A,B$ and $C$ are equivalent.

Answer 3

A,B,C are equavelent i.e. $P\rightarrow Q \equiv \sim P \vee Q$

A and C are equal because if $\rightarrow$ is true then Contradiction always true.

Comments

D is not equivalent

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Already done

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How D?

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i was taken as  $\sim$(Q $\rightarrow$ P)

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That is also not true.

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I know that sir Or wil changes to And. thats why changed that.

Answer 4

Let P and Q be statements.

1. (P→Q)⇔(¬P V Q),

2. (P→Q)⇔(¬Q→¬P), that is, the implication P→Q is logically equivalent to the contrapositive ¬Q→¬P.

Hence 1 2 and 3 are equivalent.