‘If... Then...’ Sentences
Before studying this section, you may want to review: Practice with the mathematical words ‘and’, ‘or’, and ‘is equivalent to’
One of the most common sentence structures in mathematics is ‘If $\,A\,,$ then $\,B\,$’.
This type of sentence is used in English, too: for example, ‘If it's raining, then the ground is wet.’ However, this sentence type is much more important in mathematics.
Sentences of the form ‘If $\,A\,,$ then $\,B\,$’ are called conditional sentences or implications.
Because this sentence type is so important, there are many different ways to say the same thing!
The following are equivalent: that is, if one sentence is true, then every sentence is true; and if one sentence is false, then every sentence is false.
In all these sentences, $\,A\,$ is called the hypothesis and $\,B\,$ is called the conclusion.
| If $\,A\,,$ then $\,B\,$ | Be sure that every if has a then! |
| $\,B\,,$ if $\,A\,$ | Some people state the conclusion first, to give it emphasis. |
| $\,A\,$ implies $\,B\,$ | |
| $\,A\Rightarrow B\,$ | Read this as: ‘$\,A\,$ implies $\,B\,$’ |
| Whenever $\,A\,,$ $\,B\,$ | Some people prefer the word whenever to the word if . If you use the word whenever then it is conventional to leave out the word then. |
| $\,B\,,$ whenever $\,A\,$ | Some people state the conclusion first, to give it emphasis. |
| $\,A\,$ is sufficient for $\,B\,$ |
You will see in the next section that ‘If $\,A\,,$ then $\,B\,$’ is not equivalent to ‘If $\,B\,,$ then $\,A\,$’. Therefore, the positions of $\,A\,$ and $\,B\,$ in these sentences is important. Be careful about this.
The sentence ‘If $\,A\,,$ then $\,B\,$’ is a compound sentence:
$\,A\,$ is a sentence, which can be true or false;
$\,B\,$ is a sentence, which can be true or false;
the truth of the compound sentence ‘If $\,A\,,$ then
$\,B\,$’ depends
on the truth of its subsentences $\,A\,$ and $\,B\,.$
To define a compound sentence, we must state its truth (true or false) for all possible combinations of its subsentences, and this is done by using a truth table:
| Hypothesis $\,A\,$ |
Conclusion $\,B\,$ |
Implication If $\,A\,,$ then $\,B\,$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The rows of the truth table are always written in the order given in this table.
Here are some important observations from the truth table:
- (line 1) If the hypothesis is true, and the conclusion is true, then the implication is true.
- (line 2) The only time an implication is false is when the hypothesis is true, but the conclusion is false.
- (lines 3 and 4) If the hypothesis is false, then the implication is true, regardless of the truth of the conclusion; it is said to be vacuously true in this situation.
Lines 3 and 4 are usually hardest for beginning students of logic to understand, so I like to use this analogy:
Suppose your parents have said to you,
‘If you get a $\,90\,$ or above in
AP Calculus,
then we'll buy you a car.’
Now, suppose they are telling the truth (that is, suppose the implication is true):
- If you get a $\,90\,$ or above, then they must buy you a car. (line 1 of the truth table)
-
Suppose, however, that you earn
a grade less than $\,90\,.$
(lines 3 or 4)
They could still choose to buy you a car, since they know how hard you worked. (line 3)
Or, they could put this money towards college instead, and not buy the car. (line 4)
To prove that a given implication is always true, you need to verify that line 2 of the truth table can never occur. Thus, you want to show that whenever the hypothesis is true, the conclusion must also be true.
This approach is called a direct proof of the implication:
- Assume that the hypothesis is true.
- Under this assumption, verify that the conclusion is true.
(There are other types of proofs, which will be discussed in future sections.)