Test by Bikram | Mock GATE | Test 1 | Question: 48

Question

State whether the following statements are true or false?

1. $(P\to Q) \rightarrow (Q\to P)$ always holds, for all propositions $P, Q$.
2. $\left ( \left ( P\vee Q \right )\rightarrow Q \right )$ $\rightarrow$ $\left ( Q\to \left ( P\vee Q \right ) \right )$ always holds, for all propositions $P, Q$.

• A.    $a$ is true, $b$ is false
• B.    Both $a$ and $b$ are true
• C.    $a$ is false, $b$ is true.
• D.    Both $a$ and $b$ are false.

Comments

>= should be made --> it just create confusion

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Is $\geq$ a typo?

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it doesn't look like a typo, it's intentional.

Answer 1

The statement a becomes false when P is 0 & Q is 1.

The statement b is true because

= (( P v Q) => Q) => (Q => (P v Q))

= 1

Comments

1st statement is false for P=1 and Q=0 not P=0 and Q=1.

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1) (P≥Q)→(Q≥P)

(P ->Q) -> (Q->P) = ~(~PVQ)V(~QVP)=(P/\~Q)V(~QVP)

if P =0, Q=1

(1.(~1))+((~1)+0) = 0+0 = 0(False)

2)  ((P∨Q)→Q) → (Q≥(P∨Q)) => True   ( P= 0, Q=1)

Answer 2

  Here >= is Implication

1) (P≥Q)→(Q≥P)

(P ->Q) -> (Q->P) = ~(~PVQ)V(~QVP)=(P/\~Q)V(~QVP)

if P =0, Q=1

(1.(~1))+((~1)+0) = 0+0 = 0(False)

2)  ((P∨Q)→Q) → (Q≥(P∨Q)) => alwaysTrue   ( P= 0, Q=1