Math Problem Statement

Which expressions are equivalent to p ∨ q? (Select all equivalent expressions.)

Solution

The question asks which expressions are equivalent to pqp \lor q (which reads as "p or q"). We can analyze each option using propositional logic equivalences.

Let's evaluate each expression:

  1. q¬pq \rightarrow \neg p: This means "if q, then not p." This is not logically equivalent to pqp \lor q, because pqp \lor q is true when either pp or qq is true, whereas q¬pq \rightarrow \neg p doesn't capture that truth table.

  2. qpq \lor p: This is exactly the same as pqp \lor q because disjunction (OR) is commutative. So, this is equivalent to pqp \lor q.

  3. pqp \rightarrow q: This means "if p, then q." This is not equivalent to pqp \lor q. The truth tables differ significantly.

  4. ¬qp\neg q \rightarrow p: This is logically equivalent to pqp \lor q because of the implication equivalence ¬qppq\neg q \rightarrow p \equiv p \lor q. So, this is an equivalent expression.

  5. ¬pq\neg p \rightarrow q: This is logically equivalent to pqp \lor q by the same reasoning as above (using the contrapositive form). So, this is also equivalent.

  6. qpq \land p: This is "q and p," which is not equivalent to pqp \lor q because pqp \land q is true only if both pp and qq are true, unlike pqp \lor q.

  7. ¬p¬q\neg p \land \neg q: This is the negation of pqp \lor q (using De Morgan’s law). It’s the exact opposite of pqp \lor q, so this is not equivalent.

Correct answers:

  • qpq \lor p
  • ¬qp\neg q \rightarrow p
  • ¬pq\neg p \rightarrow q

Would you like further details or have any additional questions?


Related Questions:

  1. What is the truth table for pqp \lor q?
  2. How do De Morgan's laws relate to conjunction and disjunction?
  3. What are the key differences between implication and disjunction?
  4. How does contraposition work in propositional logic?
  5. Can you provide the truth table for ¬p¬q\neg p \land \neg q?

Tip: Implication (\rightarrow) can often be rewritten as a disjunction using the equivalence pq¬pqp \rightarrow q \equiv \neg p \lor q.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Logical Equivalences
Disjunction

Formulas

Implication equivalence: ¬q → p ≡ p ∨ q
De Morgan's Law

Theorems

Implication and Disjunction Equivalency
De Morgan's Laws
Contraposition

Suitable Grade Level

University level - Logic or Discrete Mathematics