Which of the following is logically equivalent to P → Q?
AQ → P
B¬P → ¬Q
C¬P ∨ Q
DP ∧ Q
The truth table for ¬P ∨ Q matches the truth table for P → Q in every row. When P is true, ¬P is false, so ¬P ∨ Q depends entirely on Q — matching P → Q. When P is false, ¬P is true, so ¬P ∨ Q is true — matching the vacuous truth of P → Q. Option A is the converse (not equivalent). Option B is the inverse (not equivalent). Option D is conjunction, which is false whenever either is false.
Question 2 True / False
If two statements have the same truth value for three out of four rows in their truth tables but differ in one row, they are logically equivalent.
TTrue
FFalse
Answer: False
Logical equivalence requires identical truth values in EVERY row. Even one row of disagreement means the statements are not equivalent. There exists a scenario — the one corresponding to the differing row — where one statement is true and the other is false. Equivalence is an all-or-nothing property.
Question 3 Short Answer
Use De Morgan's Law to write an expression equivalent to ¬(P ∨ Q), and verify with a truth table.
Think about your answer, then reveal below.
Model answer: By De Morgan's Law, ¬(P ∨ Q) ≡ ¬P ∧ ¬Q. Verification: P=T,Q=T: ¬(T∨T)=¬T=F; ¬T∧¬T=F∧F=F. P=T,Q=F: ¬(T∨F)=¬T=F; ¬T∧¬F=F∧T=F. P=F,Q=T: ¬(F∨T)=¬T=F; ¬F∧¬T=T∧F=F. P=F,Q=F: ¬(F∨F)=¬F=T; ¬F∧¬F=T∧T=T. All rows match.
De Morgan's Laws are among the most useful equivalences in logic. ¬(P ∨ Q) ≡ ¬P ∧ ¬Q says 'not (P or Q)' is the same as 'not P and not Q.' The companion law is ¬(P ∧ Q) ≡ ¬P ∨ ¬Q. Negation distributes over connectives by swapping AND with OR.