Which of the expressions are logically equivalent to:

¬(p⟶¬(q∨p))

Question

Which of the expressions are logically equivalent to:

¬(p⟶¬(q∨p))

BlackTom AI · Accepted Answer

The question asks for expressions logically equivalent to ¬(p→¬(q∨p)).
First, translate the inner implication: p→¬(q∨p) is equivalent to ¬p ∨ ¬(q∨p).
Next simplify ¬(q∨p) using De Morgan: ¬(q∨p) = ¬q ∧ ¬p. Substituting gives ¬p ∨ (¬q ∧ ¬p).
Now apply the absorption law: ¬p ∨ (¬q ∧ ¬p) = ¬p.
Therefore p→¬(q∨p) ≡ ¬p, and the original expression ¬(p→¬(q∨p)) becomes ¬(¬p) ≡ p.
Turning to the provided option, and noting it is the only candidate given: ¬(¬p ∧ (¬p ∨ ¬q)).
Use De Morgan carefully: this equals ¬(¬p) ∨ ¬(¬p ∨ ¬q) = p ∨ (p ∧ q).
Since p ∨ (p ∧ q) simplifies to p (absorption), this option also reduces to p.
Thus, the expression ¬(¬p ∧ (¬p ∨ ¬q)) is indeed logically equivalent to the original ¬(p→¬(q∨p)).
In general, to verify equivalence, one can perform step-by-step boolean algebra: convert implications, apply De Morgan’s laws, and use absorption and distributive properties as needed to arrive at the same simplified form. This approach helps reveal common misconceptions, such as assuming the negation of an implication behaves like the negation of the antecedent alone, which the step-by-step simplification shows is not the case.
If there were other options, each would need its own equivalent simplification path to check against p or any final simplified form; without additional options, we focus on demonstrating how the given option reduces to p and thus matches the original expression's truth function.

Section Merge: Discrete Math/Cs (PCMPSC360-2025SP) Quiz 1: Logic and sets. Graded.

Which of the expressions are logically equivalent to: ¬(p⟶¬(q∨p))

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