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))

View Explanation

Log in for full answers

Similar Questions

Suppose that P is a true statement, and Q is a false statement. What is the truth value of the following statement (¬(P⟹Q))∧(P∨Q)

Consider the following logical statement:[math: (P⇒Q)∧¬Q](P \Rightarrow Q) \land \neg QWhich of the following is true about this statement?

Suppose that 𝑊 , 𝑊 1 and 𝑊 2 are wffs in the Propositional Calculus, such that the following sequent is valid: ⊢ ∼ 𝑊 1 ⇒ ∼ 𝑊 2 Which one of the following sequents is valid?

Here is an unfinished proof of the following sequent: 𝑃 ∧ ( 𝑄 ∨ 𝑅 ) ⊢ ( 𝑃 ∧ 𝑄 ) ∨ ( 𝑃 ∧ 𝑅 ) Which one of the following would become a correct line (10) that makes the proof complete?

Consider the following truth table for a wff 𝑊 formed using propositional variables 𝑃 , 𝑄 and 𝑅 : Which one of the following wffs would be correct for 𝑊 ?

Consider the following truth table: Which one of the following wffs correctly represents 𝑊 ?

Which one of the following sequents is valid?

Which one of the following is a valid sequent in the Propositional Calculus:

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler