سوال ۱۹

کدام یک از عبارات منطقی زیر معادل با عبارت $\small(B \wedge \sim D) \vee (\sim C \wedge D)$ است؟

  1. $\small(B \vee \sim C) \wedge (\sim D \vee D)$
  2. $\small(B \vee \sim D) \wedge (C \vee D)$
  3. $\small(B \vee D) \wedge (\sim C \vee \sim D)$
  4. $\small(B \vee \sim D) \wedge (\sim C \vee D)$
  5. $\small(B \vee D) \wedge (C \vee \sim D)$

پاسخ

گزینه (۳) درست است.

$(B\wedge \sim D) \vee (\sim C \wedge D)
\equiv [B \vee (\sim C \wedge D)] \wedge [\sim D \vee ( \sim C \wedge D)]
\equiv [(B \vee \sim C) \wedge (B \vee D) \wedge ( \sim D \vee \sim C) \wedge (\sim D \vee D)]
\equiv (B \vee \sim C) \wedge (\sim D \vee \sim C) \wedge (B \vee D)
\equiv [ \sim C \vee ( B \wedge \sim D)] \wedge (B \vee D)
\equiv [\sim C \wedge (B \vee D)] \vee [( B \wedge \sim D) \wedge (B \vee D)]
\equiv [\sim C \wedge (B \vee D)] \vee [(\sim D \wedge B \wedge (B \vee D)]
\equiv [\sim C \wedge (B \vee D)] \vee [(\sim D \wedge B) \vee F
\equiv [\sim C \wedge (B \vee D)] \vee [(\sim D \wedge B) \vee (\sim D \wedge D)]
\equiv [\sim C \wedge (B \vee D)] \vee [\sim D \wedge (B \vee D)]
\equiv \small(B \vee D) \wedge (\sim C \vee \sim D)$