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10 The Nuts and Bolts of Proof, Third Edition
Possibilities
^
A is true
^
A is false
y
^
B is true
B is false
B is true
B is false
Thus, we can consider four possibilities:
1. A is true and B is true.
2. A is true and B is false.
3. A is false and B is true.
4. A is false and B is false.
Case 1. You do go home and you do take your parents out to dinner.
Your statement is true.
Case 2. You go home for the weekend, but you do not take your parents
out to dinner. You have been caught lying! Your statement is false.
Cases 3 and 4. You cannot be accused of lying if you did not go home,
but you did take your parents out to dinner, because they came to visit. If
you did not go home, nobody can accuse you of lying if you did not take
your parents out to dinner. It is very important to notice that you had
not specified what you would do in case you were not going home (A is false).
So, whether you did take your parents out to dinner or not, you did not lie.
In conclusion, there is only one case in which your statement is false—
namely, when A is true and B is false. This is a general feature of statements
of the form "If A, then B" or "A implies B."
A statement of the form "If A, then B" is true if we can prove that it is
impossible for A to be true and B to be false at the same time; that is,
whenever A is true, B must be true as well.
The statement "If A, then B" can be reworded as "A is a sufficient
condition for B" and as "B is a necessary condition for A." The mathematical
use of the words "sufficient" and "necessary" is very similar to their everyday
use. If a given statement is true and it provides enough (sufficient)
information to reach the conclusion, then the statement is called a sufficient
condition. If a statement is an inevitable (certain) consequence of a given
statement, it is called a necessary condition. A condition can be sufficient but
not necessary or necessary but not sufficient.
As an example, consider the statement "If an animal is a cow, then it has
four legs." Having four legs is a necessary condition for an animal to be a
cow, but it is not a sufficient condition for identifying a cow, as it is possible