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:
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:
(There are other types of proofs, which will be discussed in future sections.)
|
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
|