Let P- Q- and R be three atomic propositional assertions - Let X denote P - Q - R and Y denote P-R - Q-R- Which one of the following is a tautology-

X: (P ∨ Q) → R nbsp; Y: (P→ R) ∨ (Q→ R) X: P + Q → R ≡ (P + Q)' + R ≡ P'Q' + R Y:(P' + R) + (Q' + R) ≡ P' + Q' + R Clearly X ≠ Y Consider X → Y ≡ (P'Q' + R)→ ...

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Byju's Answer

Standard VIII

Mathematics

Properties of Negative Numbers

Let P, Q, and...

Question

Let P, Q, and R be three atomic propositional assertions . Let X denote (P $\lor$ Q) $\to$ R and Y denote ( $P \to$ R) $\lor$ ( $Q \to$ R). Which one of the following is a tautology?

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Solution

The correct option is B X $\to$ Y
$X : (P \lor Q) \to R$
$Y : (P \to R) \lor (Q \to R)$
$X : P + Q \to R \equiv (P + Q)^{'} + R \equiv P^{'} Q^{'} + R$
$Y : (P^{'} + R) + (Q^{'} + R) \equiv P^{'} + Q^{'} + R$
Clearly X $\neq$ Y
Consider X $\to$ Y
$\equiv (P^{'} Q^{'} + R) \to (P^{'} + Q^{'} + R)$
$\equiv (P^{'} Q^{'} + R) + P^{'} + Q^{'} + R$
$\equiv (P^{'} Q^{'})^{'} \cdot R^{'} + P^{'} + Q^{'} + R$
$\equiv (P + Q) \cdot R^{'} + P^{'} + Q^{'} + R$
$\equiv P R^{'} + Q R^{'} + P^{'} + Q^{'} + R$
$\equiv (P R^{'} + R) + (Q R^{'} + Q^{'}) + P^{'}$
$\equiv (P + R) (R^{'} + R) + (Q + Q^{'}) \times (R^{'} + Q^{'}) + P^{'}$
$\equiv (P + R) + (R^{'} + Q^{'}) + P^{'}$
$\equiv P + P^{'} + R + R^{'} + Q^{'}$
$\equiv 1 + 1 + Q^{'}$
$\equiv 1$
$∴ X \to Y$ is a tautology.

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Let P, Q, and R be three atomic propositional assertions . Let X denote (P ∨ Q) → R and Y denote (P→R) ∨ (Q→R). Which one of the following is a tautology?

Let P, Q, and R be three atomic propositional assertions . Let X denote (P $\lor$ Q) $\to$ R and Y denote ( $P \to$ R) $\lor$ ( $Q \to$ R). Which one of the following is a tautology?