← Logic

Logic

Converse, Inverse, Contrapositive

Lesson

From an original conditional p→qp \to q, three related conditionals fall out.

The three variants

  • Converse: q→pq \to p. Swap the parts.
  • Inverse: ¬p→¬q\neg p \to \neg q. Negate the parts.
  • Contrapositive: ¬q→¬p\neg q \to \neg p. Swap AND negate.

Equivalence facts

  • Original ≡ Contrapositive (always).
  • Converse ≡ Inverse (always).
  • Original ≢ Converse (and ≢ Inverse) in general.

How to type your answer

For classification: 1 converse, 2 inverse, 3 contrapositive, 4 original. For yes/no: 1 = yes, 0 = no.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Original p→q. Given: q→p\text{Original } p \to q. \text{ Given: } q \to p

Problem 2

Original p→q. Given: ¬p→¬q\text{Original } p \to q. \text{ Given: } \neg p \to \neg q

Problem 3

Original p→q. Given: ¬q→¬p\text{Original } p \to q. \text{ Given: } \neg q \to \neg p

Problem 4

Which is logically equivalent to p→q?\text{Which is logically equivalent to } p \to q?

Practice

Standard problems matching the lesson.

Problem 5

Original r→s. Given: s→r\text{Original } r \to s. \text{ Given: } s \to r

Problem 6

Original r→s. Given: ¬r→¬s\text{Original } r \to s. \text{ Given: } \neg r \to \neg s

Problem 7

Original r→s. Given: ¬s→¬r\text{Original } r \to s. \text{ Given: } \neg s \to \neg r

Problem 8

Original a→b. Given: a→b\text{Original } a \to b. \text{ Given: } a \to b

Problem 9

Converse and inverse are equivalent?\text{Converse and inverse are equivalent?}

Problem 10

Original and contrapositive are equivalent?\text{Original and contrapositive are equivalent?}

Problem 11

Original and converse are equivalent?\text{Original and converse are equivalent?}

Problem 12

Original and inverse are equivalent?\text{Original and inverse are equivalent?}

Problem 13

“If even, then div by 2.” “If div by 2, then even”\text{``If even, then div by 2.'' ``If div by 2, then even''}

Problem 14

“If even, then div by 2.” “If not even, then not div by 2”\text{``If even, then div by 2.'' ``If not even, then not div by 2''}

Problem 15

“If even, then div by 2.” “If not div by 2, then not even”\text{``If even, then div by 2.'' ``If not div by 2, then not even''}

Problem 16

“If rains, ground wet.” “If ground wet, rains”\text{``If rains, ground wet.'' ``If ground wet, rains''}

Problem 17

“If rains, ground wet.” “If ground not wet, no rain”\text{``If rains, ground wet.'' ``If ground not wet, no rain''}

Problem 18

“If square, then 4 sides.” Equivalent form (contrapositive)?\text{``If square, then 4 sides.'' Equivalent form (contrapositive)?}

Challenge

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

Problem 19

Original p→q. Given: ¬¬p→¬¬q\text{Original } p \to q. \text{ Given: } \neg\neg p \to \neg\neg q

Problem 20

The converse of the inverse of p→q\text{The converse of the inverse of } p \to q

Problem 21

The contrapositive of the contrapositive\text{The contrapositive of the contrapositive}

Problem 22

Inverse of the converse\text{Inverse of the converse}

Problem 23

A statement and its converse always have the same truth value?\text{A statement and its converse always have the same truth value?}

Problem 24

A statement and its contrapositive always have the same truth value?\text{A statement and its contrapositive always have the same truth value?}

Problem 25

Inverse of inverse\text{Inverse of inverse}

Ask the tutor

Stuck on a concept? Want another example? Ask anything about this topic.

Type your own question below, or tap one of the suggestions. The tutor can re-explain the lesson, work through a specific problem with you, generate fresh practice tuned to where you are, or check your reasoning.

Quiz

Test yourself on this topic →

10 questions, no hints. About 5 minutes.