‘IF ... THEN ...’ SENTENCES
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)
• Before studying this section, you may want to review:

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:

DEFINITION implication
 hypothesis$\,A\,$ conclusion$\,B\,$ implicationIf $\,A\,$, 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.)

Master the ideas from this section