A) \[(\tilde{\ }p)\wedge (\tilde{\ }q)\]
B) \[p\wedge (\tilde{\ }q)\]
C) \[p\wedge q\]
D) \[p\vee (\tilde{\ }q)\]
Correct Answer: A
Solution :
\[((p\wedge q)\vee (p\vee \tilde{\ }q))\wedge (\tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv ((p\vee (p\vee \tilde{\ }q))\wedge (q\vee \tilde{\ }q)))\wedge (\tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv ((p\vee \tilde{\ }q)\wedge (q\vee \tilde{\ }q\vee p))\wedge (\tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv ((p\vee \tilde{\ }q)\wedge (t\vee p))\wedge (\tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv ((p\vee \tilde{\ }q)\wedge t)\wedge (\tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv (p\vee \tilde{\ }q)\wedge (\tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv (p\wedge \tilde{\ }p\wedge \tilde{\ }q)\vee (\tilde{\ }q\wedge \tilde{\ }p\wedge \tilde{\ }q)\] \[\equiv (c\wedge \tilde{\ }q)\vee (\tilde{\ }q\wedge \tilde{\ }p)\] \[\equiv c\vee (\tilde{\ }q\wedge \tilde{\ }p)\] \[\equiv \tilde{\ }p\wedge \tilde{\ }q\]
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