# If q is false and $p\wedge q\leftrightarrow r$  is true, then which one of the following statements is a tautology?Option 1)  $(p \vee r)\rightarrow (p\wedge r)$Option 2)  $p\wedge r$Option 3)  $p \vee r$Option 4)  $(p \wedge r)\rightarrow (p\vee r)$

Tautology -

A statement pattern is called tautalogy, if it is always true, whatever may be the truth values of constitute statements.

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Truth Table of 'AND' operator -

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Truth Table of "if and only if" -

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Truth table of 'OR' operator -

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$(p\wedge q)\leftrightarrow r$ is true

Case 1  $p\wedge q$ is true and r is true It is not possible as q is false

Case 2    $p\wedge q$ is false and r is false

$\Rightarrow p\equiv T/F$

$r\equiv F$

$p\vee r\rightarrow p\wedge r\Rightarrow T or F \rightarrow F$ which may be T/F

$p\wedge r\rightarrow p \vee r\Rightarrow F \rightarrow T/F$ which is always True.

Option 1)

$(p \vee r)\rightarrow (p\wedge r)$

Option 2)

$p\wedge r$

Option 3)

$p \vee r$

Option 4)

$(p \wedge r)\rightarrow (p\vee r)$

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