Page 65 - HOW TO PROVE IT: A Structured Approach, Second Edition

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The Conditional and Biconditional Connectives 51
Some of these may require further explanation. The second expression,
“Q,if P,” is just a slight rearrangement of the statement “If P then Q,” so
it should make sense that it means P → Q. As an example of a statement of
the form “P only if Q,” consider the sentence “You can run for president only if
you are a citizen.” In this case, P is “You can run for president” and Q is “You
are a citizen.” What the statement means is that if you’re not a citizen, then you
can’t run for president, or in other words ¬Q →¬P. But by the contrapositive
law, this is equivalent to P → Q.
Think of “P is a sufficient condition for Q” as meaning “The truth of P
suffices to guarantee the truth of Q,” and it should make sense that this should
be represented by P → Q. Finally, “Q is a necessary condition for P” means
that in order for P to be true, it is necessary for Q to be true also. This means
that if Q isn’t true, then P can’t be true either, or in other words, ¬Q →¬P.
Once again, by the contrapositive law we get P → Q.

Example 1.5.3. Analyze the logical forms of the following statements:

1. If at least ten people are there, then the lecture will be given.
2. The lecture will be given only if at least ten people are there.
3. The lecture will be given if at least ten people are there.
4. Having at least ten people there is a sufficient condition for the lecture being
given.
5. Having at least ten people there is a necessary condition for the lecture being
given.

Solutions
Let T stand for the statement “At least ten people are there” and L for “The
lecture will be given.”
1. T → L.
2. L → T . The given statement means that if there are not at least ten people
there, then the lecture will not be given, or in other words ¬T →¬L.By
the contrapositive law, this is equivalent to L → T .
3. T → L. This is just a rephrasing of statement 1.
4. T → L. The statement says that having at least ten people there suffices to
guarantee that the lecture will be given, and this means that if there are at
least ten people there, then the lecture will be given.
5. L → T . This statement means the same thing as statement 2: If there are
not at least ten people there, then the lecture will not be given.