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Logic

Logical Equivalence

Lesson

Two statements are logically equivalent if they have the same truth value in every row of their truth tables. Symbol: ≡\equiv.

Famous equivalences

  • Commutative: p∧q≡q∧pp \wedge q \equiv q \wedge p, same for ∨.
  • Double negation: ¬¬p≡p\neg\neg p \equiv p.
  • De Morgan: ¬(p∧q)≡¬p∨¬q\neg(p \wedge q) \equiv \neg p \vee \neg q.
  • Conditional as OR: p→q≡¬p∨qp \to q \equiv \neg p \vee q.
  • Contrapositive: p→q≡¬q→¬pp \to q \equiv \neg q \to \neg p.
  • Biconditional: p↔q≡(p→q)∧(q→p)p \leftrightarrow q \equiv (p \to q) \wedge (q \to p).

Common NON-equivalences

  • p→qp \to q is NOT equivalent to q→pq \to p (converse).
  • p→qp \to q is NOT equivalent to ¬p→¬q\neg p \to \neg q (inverse).

Type 1 if the two statements are equivalent, 0 if not.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

p∧q ≡? q∧pp \wedge q \ \stackrel{?}{\equiv}\ q \wedge p

Problem 2

p∨q ≡? q∨pp \vee q \ \stackrel{?}{\equiv}\ q \vee p

Problem 3

p→q ≡? q→pp \to q \ \stackrel{?}{\equiv}\ q \to p

Problem 4

¬¬p ≡? p\neg\neg p \ \stackrel{?}{\equiv}\ p

Practice

Standard problems matching the lesson.

Problem 5

¬(p∧q) ≡? ¬p∨¬q\neg(p \wedge q) \ \stackrel{?}{\equiv}\ \neg p \vee \neg q

Problem 6

¬(p∨q) ≡? ¬p∧¬q\neg(p \vee q) \ \stackrel{?}{\equiv}\ \neg p \wedge \neg q

Problem 7

p→q ≡? ¬p∨qp \to q \ \stackrel{?}{\equiv}\ \neg p \vee q

Problem 8

p→q ≡? ¬q→¬pp \to q \ \stackrel{?}{\equiv}\ \neg q \to \neg p

Problem 9

p↔q ≡? (p→q)∧(q→p)p \leftrightarrow q \ \stackrel{?}{\equiv}\ (p \to q) \wedge (q \to p)

Problem 10

p∧p ≡? pp \wedge p \ \stackrel{?}{\equiv}\ p

Problem 11

p∨p ≡? pp \vee p \ \stackrel{?}{\equiv}\ p

Problem 12

p∧F ≡? Fp \wedge F \ \stackrel{?}{\equiv}\ F

Problem 13

p∨T ≡? Tp \vee T \ \stackrel{?}{\equiv}\ T

Problem 14

p∧T ≡? pp \wedge T \ \stackrel{?}{\equiv}\ p

Problem 15

p∨F ≡? pp \vee F \ \stackrel{?}{\equiv}\ p

Problem 16

p→q ≡? p∧¬qp \to q \ \stackrel{?}{\equiv}\ p \wedge \neg q

Problem 17

¬(p→q) ≡? p∧¬q\neg(p \to q) \ \stackrel{?}{\equiv}\ p \wedge \neg q

Problem 18

p∧(q∨r) ≡? (p∧q)∨(p∧r)p \wedge (q \vee r) \ \stackrel{?}{\equiv}\ (p \wedge q) \vee (p \wedge r)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

p∨(q∧r) ≡? (p∨q)∧(p∨r)p \vee (q \wedge r) \ \stackrel{?}{\equiv}\ (p \vee q) \wedge (p \vee r)

Problem 20

p→q ≡? ¬p→¬qp \to q \ \stackrel{?}{\equiv}\ \neg p \to \neg q

Problem 21

¬(p∧q) ≡? ¬p∧¬q\neg(p \wedge q) \ \stackrel{?}{\equiv}\ \neg p \wedge \neg q

Problem 22

¬(p∨q) ≡? ¬p∨¬q\neg(p \vee q) \ \stackrel{?}{\equiv}\ \neg p \vee \neg q

Problem 23

p∧(p∨q) ≡? pp \wedge (p \vee q) \ \stackrel{?}{\equiv}\ p

Problem 24

p∨(p∧q) ≡? pp \vee (p \wedge q) \ \stackrel{?}{\equiv}\ p

Problem 25

p↔q ≡? ¬p↔¬qp \leftrightarrow q \ \stackrel{?}{\equiv}\ \neg p \leftrightarrow \neg q

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