Consider the following expressions: I. False II. Q III. True IV. P ∨ Q V. P ∧…
Consider the following expressions:
I. False
II. Q
III. True
IV. P ∨ Q
V. P ∧ Q
The number of expressions given above that are not logically equivalent to P ∧ (P → Q) is ______.
Answer: 4 — ConceptTwo propositions are logically equivalent when they take the same truth value for every combination of truth values of their variables (identical truth…
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Correct answer: 4
Concept
Two propositions are logically equivalent when they take the same truth value for every combination of truth values of their variables (identical truth tables). The material-implication law rewrites an implication in terms of negation and disjunction: P → Q ≡ ¬P ∨ Q. Combined with the distributive law (A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C)) and the complement law (A ∧ ¬A ≡ False), this lets a conjunction-with-implication be reduced to a simpler equivalent form.
Application
Reduce P ∧ (P → Q) step by step:
Rewrite the implication using the material-implication law: P ∧ (P → Q) ≡ P ∧ (¬P ∨ Q).
Distribute the conjunction over the disjunction: P ∧ (¬P ∨ Q) ≡ (P ∧ ¬P) ∨ (P ∧ Q).
Apply the complement law: P ∧ ¬P is always False, so the expression becomes False ∨ (P ∧ Q).
Apply the identity law for disjunction with False: False ∨ (P ∧ Q) ≡ P ∧ Q.
So P ∧ (P → Q) reduces to P ∧ Q, which is exactly expression V.
Cross-check — truth table
Independently verifying against a full truth table confirms the algebraic reduction and lets every expression be compared row by row:
P | Q | P ∧ (P → Q) | I. False | II. Q | III. True | IV. P ∨ Q | V. P ∧ Q |
|---|---|---|---|---|---|---|---|
T | T | T | F | T | T | T | T |
T | F | F | F | F | T | T | F |
F | T | F | F | T | T | T | F |
F | F | F | F | F | T | F | F |
Reading down each column against the target column P ∧ (P → Q): False, Q, True, and P ∨ Q each disagree with the target in at least one row, so none of them is logically equivalent to it. Only P ∧ Q matches the target row for row, confirming the algebraic result.
Result
Expressions I (False), II (Q), III (True), and IV (P ∨ Q) are not logically equivalent to P ∧ (P → Q); only expression V (P ∧ Q) is. So the count of expressions that are NOT logically equivalent to P ∧ (P → Q) is 4.