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A conditional statement combines two statements: a hypothesis p and a conclusion q. Every conditional statement can be written in if-then form.
rc Words: & Ifp, then q. Symbols: & p ⇒ q
The hypothesis comes after the word if
and the conclusion after the word then.
The symbolic form is read as p implies q.
If a figure is a square_(hypothesis p),
thenit has four sides_(conclusion q).
A conditional statement is not always true. Its truth depends on the truth values of p and q, and a truth table lists every possible combination.
| Conditional Statement | ||
|---|---|---|
| p | q | p ⇒ q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Only one row makes the statement false: the one where the hypothesis is true and the conclusion is false. Every other combination leaves it true.
In the first scenario, the hypothesis p, a figure is a square,
is true and the conclusion q, it has four sides,
is also true. If a figure is a square_T, thenit has four sides_T. ✓
When both parts are true, the conditional statement is true as well.
Here the hypothesis is true but the conclusion is false, which describes a square without four sides. If a figure is a square_T, thenit does not have four sides_F. * A true hypothesis followed by a false conclusion is the one combination that breaks a conditional statement.
Now the hypothesis is false and the conclusion is true, which describes a figure that is not a square but does have four sides, such as a rectangle. If a figure is not a square_F, thenit has four sides_T. ✓ The original statement promises nothing about figures that are not squares, so a rectangle does not contradict it. The statement remains true.
In the last scenario, both parts are false, which describes a figure that is neither a square nor four-sided, such as a triangle. If a figure is not a square_F, thenit does not have four sides_F. ✓ Again, a false hypothesis makes the conclusion irrelevant, so nothing is contradicted.