‘If... Then...’ Sentences

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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.

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:

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 by practicing below:

When you're done practicing, move on to:

Contrapositive and Converse