Question

Use any of the two proof methods to prove:

((~a^b)^(b^c))^~b

Answer 1

Direct Proof:

$((\neg a\land b)\land(b\land c))\land \neg b\equiv \neg a\land b\land c \land \neg b\equiv \neg a \land c \land 0\equiv 0$

Proof by Contrapositive:

$\neg(((\neg a\land b)\land(b\land c))\land \neg b)\equiv \neg (\neg a \land c \land 0)\equiv \neg0\equiv 1$

so,

$((\neg a\land b)\land(b\land c))\land \neg b\equiv 0$