Lesson 10 of 1524
Logic
A statement is a sentence that is true or false, and a conditional fails in only one case.
Practice this chapter“And” is true only when both parts are true. “Or,” in this book, is true when at least one part is true. A conditional “if p, then q” is false only when p is true and q is false.
The contrapositive of “if p, then q” is “if not q, then not p.” It has the same truth as the original. The converse, “if q, then p,” does not.
When a conditional is false
p → q is false only for true → false
A true hypothesis with a false conclusion is the only failure.
Worked example
“If it rains, the ground is wet.” What is the contrapositive?
- 1Name the parts. p is “it rains.” q is “the ground is wet.” The statement is if p, then q.
- 2The contrapositive swaps the parts and negates both of them: if not q, then not p.
- 3Not q is “the ground is not wet.” Not p is “it did not rain.”
- 4Put those together: if the ground is not wet, then it did not rain.
- 5The converse only swaps the parts: “if the ground is wet, then it rained.” That is a different statement and can be false while the original is true.
Result: If the ground is not wet, it did not rain.
Why. A conditional matches its contrapositive because the only way the original fails is a true p with a false q, and that is the same failure as a true “not q” with a false “not p.” Swapping without the negations does not keep that match.
The converse can be false while the original is true. A wet ground does not prove that it rained.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.