p ∧ r ⇔ p ∧ ~ q is equivalent to ~ p when r is

Question

p ∧ r ⇔ p ∧ ~ q is equivalent to ~ p when r is, p, ~ p, q, ~ q

MARKS App · Accepted Answer

Given p ∧ r ⇔ p ∧ ~ q ≡ ~ p Assuming all option one by one, Taking r = p p q ~ p ~ q p ∧ p p ∧ ~ q p ∧ r ⇔ p ∧ ~ q T T F F T F F T F F T T T T F T T F F F T F F T T F F T Now taking r = ~ p p q ~ p ~ q p ∧ ~ p p ∧ ~ q p ∧ r ⇔ p ∧ ~ q T T F F F F T T F F T F T F F T T F F F T F F T T F F T Taking r = q p q ~ p ~ q p ∧ q p ∧ ~ q p ∧ r ⇔ p ∧ ~ q T T F F T F F T F F T F T F F T T F F F T F F T T F F T So clear, p ∧ r ⇔ p ∧ ~ q ≡ ~ p

$p$	$q$	$~ p$	$~ q$	$p \land p$	$p \land ~ q$	$(p \land r) \Leftrightarrow (p \land (~ q))$
$T$	$T$	$F$	$F$	$T$	$F$	$F$
$T$	$F$	$F$	$T$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$~ p$	$~ q$	$p \land ~ p$	$p \land ~ q$	$(p \land r) \Leftrightarrow (p \land (~ q))$
$T$	$T$	$F$	$F$	$F$	$F$	$T$
$T$	$F$	$F$	$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$~ p$	$~ q$	$p \land q$	$p \land ~ q$	$(p \land r) \Leftrightarrow (p \land (~ q))$
$T$	$T$	$F$	$F$	$T$	$F$	$F$
$T$	$F$	$F$	$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

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