Which of the following statements is a tautol | vedclass

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(A) tautology is a statement that is always true for all possible truth values of its components.Let us simplify the expression $S = p \wedge ((q \vee

Vedclass · Accepted Answer

$\left[ {\left\{ {p \wedge \left( {\left( {q \vee t} ight) \wedge p} ight)} ight\} 	o \left\{ {\left( {q \vee r} ight) \wedge \left( {p \vee t} ight)} ight\}} ight] \leftrightarrow \left[ { \sim \left( {q \vee r} ight) 	o \sim p} ight]$

Vedclass · Answer

(A) tautology is a statement that is always true for all possible truth values of its components.Let us simplify the expression $S = p \wedge ((q \vee t) \wedge p)$.Since $q \vee t$ is a tautology (as $t$ is a tautology),we have $q \vee t \equiv t$.Thus,$S = p \wedge (t \wedge p) = p \wedge p = p$.Now,check option $D$: $S \leftrightarrow t$ becomes $p \leftrightarrow t$.Since $p$ can be false,$p \leftrightarrow t$ is not a tautology.Check option $A$: The expression is $[(p 	o (q \vee r) \wedge (p \vee t))] \leftrightarrow [\sim(q \vee r) 	o \sim p]$.Using $p 	o q \equiv \sim p \vee q$,the right side is $\sim(\sim(q \vee r)) \vee \sim p = (q \vee r) \vee \sim p$.The left side simplifies to $p 	o ((q \vee r) \wedge (p \vee t))$.Since $p \vee t$ is a tautology,this is $p 	o (q \vee r) \equiv \sim p \vee (q \vee r)$.Both sides are identical,hence the statement is a tautology.

Which of the following statements is a tautology?

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