If the Boolean expression (p [ ] q) [ ] (q * ([ ]p) is a tautology, then the boolean expre…

Question

If the Boolean expression (p $$ \Rightarrow $$ q) $$ \Leftrightarrow $$ (q * ($$ \sim $$p) is a tautology, then the boolean expression (p * ($$ \sim $$q)) is equivalent to :

Accepted Answer

The Boolean expression

$(p \Rightarrow q) \Leftrightarrow\left(q^*(\sim p)ight)$ is a tautology.

Making the truth table for this

p
    q
    $p ightarrow q$
    $q^* \sim p$
    $\sim q$
    $-q \wedge p$
    $\sim(\sim {q} \wedge p)$

$\mathrm{T}$
    $\mathrm{T}$
    $\mathrm{T}$
    $\mathrm{T}$
    $\mathrm{F}$
    $\mathrm{F}$
    $\mathrm{T}$

T
    F
    F
    F
    T
    T
    F

F
    T
    T
    T
    F
    F
    T

F
    F
    T
    T
    T
    F
    T

$	herefore \quad \sim(\sim q \wedge p)=q \vee \sim p=\sim p \vee q$

$	herefore{ }^*$ is equivalent to $v$.

So, $\quad p^* \sim q=p \vee \sim q=\sim q \vee p$

$=q \Rightarrow p$

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