Practice Exercise 4 IV. Write the rule of replacement in the following proof of logical equivalences. \[ \begin{array}{l} (p \rightarrow q) \wedge(p \rightarrow r) \equiv p \rightarrow(q \cap r) \\ \text { Proof: }(p \rightarrow q) \wedge(p \rightarrow r) \equiv(\neg p \vee q) \wedge(\neg p \vee r) \\ \equiv \neg p \vee(q \wedge r) \\ \equiv p \rightarrow(q \wedge \gamma) \\ \end{array} \]
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(p → q) ∧ (p → r) ≡ (¬p ∨ q) ∧ (¬p ∨ r): This step uses the rule of replacement known as Implication, which states that p → q is logically equivalent to ¬p ∨ q. Show more…
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Use truth tables to verify these equivalences. $\begin{array}{ll}{\text { a) } p \wedge \mathbf{T} \equiv p} & {\text { b) } p \vee \mathbf{F} \equiv p} \\ {\text { c) } p \wedge \mathbf{F} \equiv \mathbf{F}} & {\text { d) } p \vee \mathbf{F} \equiv \mathbf{T}} \\ {\text { e) } p \vee p \equiv p} & {\text { f) } p \wedge p \equiv p}\end{array}$
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Use truth tables to verify these equivalences: a) p ^ T ≡ p b) p ∨ F ≡ p c) p ^ F ≡ F d) p ∨ T ≡ T e) p ∨ p ≡ p f) p ^ p ≡ p Show that ¬(¬p) and p are logically equivalent. Use truth tables to verify the commutative laws a) p ∨ q ≡ q ∨ p. b) p ^ q ≡ q ^ p.
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Use truth tables to verify these equivalences: a) p ^ T ≡ p b) p ∨ F ≡ p c) p ∧ F ≡ F d) p ∨ T ≡ T e) p ∨ p ≡ p f) p ∧ p ≡ p
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