Let p be the statement 'x is an irrational nu | vedclass

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Question

(A) Given $p$: $x$ is irrational,so $\sim p$: $x$ is rational.Given $q$: $y$ is transcendental.Statement $r$ is '$x$ is rational or $y$ is transcenden

Vedclass · Accepted Answer

Statement-$1$ is false. Statement-$2$ is true.

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(A) Given $p$: $x$ is irrational,so $\sim p$: $x$ is rational.Given $q$: $y$ is transcendental.Statement $r$ is '$x$ is rational or $y$ is transcendental',which is $\sim p \lor q$.Let's evaluate the truth tables for the given statements:$p$$q$$\sim p$$\sim q$$r = (\sim p \lor q)$$q \lor p$$(p \Leftrightarrow \sim q)$$T$$T$$F$$F$$T$$T$$F$$T$$F$$F$$T$$F$$T$$T$$F$$T$$T$$F$$T$$T$$T$$F$$F$$T$$T$$T$$F$$F$Comparing the columns:$r$ is $(\sim p \lor q)$.Statement-$1$: $r \equiv q \lor p$. From the table,$r$ and $q \lor p$ are not equivalent. So,Statement-$1$ is false.Statement-$2$: $r \equiv (p \Leftrightarrow \sim q)$. From the table,$r$ and $(p \Leftrightarrow \sim q)$ are not equivalent. So,Statement-$2$ is false.Wait,re-evaluating the logic: $r$ is $\sim p \lor q$. $q \lor p$ is not $\sim p \lor q$.$(p \Leftrightarrow \sim q)$ is true when $p$ and $\sim q$ have the same truth value,i.e.,$p$ and $q$ have opposite truth values.Thus,both statements are false.

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