Connecting conditions
Distinguish and, or, and implication, and compare the converse, inverse, and contrapositive.
Meaning
Connecting conditions with and, or, or if–then creates new conditions or statements. The contrapositive of “if P, then Q” is “if not Q, then not P,” and has the same truth value as the original. An implication fails exactly when P is true and Q is false. Its contrapositive fails in exactly the same case: Q is false while P is true. Thus their truth values agree.
Example
The contrapositive of “a multiple of 4 is even” is “a number that is not even is not a multiple of 4.” The converse, “an even number is a multiple of 4,” is false because of 6. In mathematics, “or” usually includes the case where both are true. A true statement need not have a true converse or inverse.
Practice for this concept
Scope and references
Topics
And, or, implication, converse, inverse, contrapositive
Learning goal
Distinguish and, or, and implication, and compare the converse, inverse, and contrapositive.
Helpful background
Review truth, falsity, and negation.
Combining conditions
These choice exercises use specified finite sets, conditions, or proof steps. They do not establish constructing new counterexamples or proofs, or mastery of the whole logic area.