Which of the following is a contradiction?

1) (p ˄ q) ˄ ~ (p ˅ q)

2) p ˅ (~ p ˄ q)

3) (p ⇒ q) ⇒ p

4) None of these

Solution: (1) (p ˄ q) ˄ ~ (p ˅ q)

\(\begin{array}{l}\begin{array}{cccccc} p & q & p \wedge q & p \vee q & \sim(p \vee q) & (p \wedge q) \wedge \sim(p \vee q) \\ \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \therefore(p \wedge q \wedge(\sim(p \vee q)) \text { is a contradiction. } \end{array}\end{array} \)

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