Page 64 - HOW TO PROVE IT: A Structured Approach, Second Edition

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50 Sentential Logic
3. If the game has been canceled, then it’s either raining or snowing.
4. If it’s raining then the game has been canceled, and if it’s snowing then the
game has been canceled.
5. If it’s neither raining nor snowing, then the game hasn’t been canceled.

Solution

We translate all of the statements into the notation of logic, using the follow-
ing abbreviations: R stands for the statement “It’s raining,” S stands for “It’s
snowing,” and C stands for “The game has been canceled.”
1. (R ∨ S) → C.
2. ¬C → (¬R ∧¬S). By one of DeMorgan’s laws, this is equivalent to
¬C →¬(R ∨ S). This is the contrapositive of statement 1, so they are
equivalent.
3. C → (R ∨ S). This is the converse of statement 1, which is not equivalent
to it. You can verify this with a truth table, or just think about what the state-
ments mean. Statement 1 says that rain or snow would result in cancelation
of the game. Statement 3 says that these are the only circumstances in which
the game will be canceled.
4. (R → C) ∧ (S → C). This is also equivalent to statement 1, as the following
reasoning shows:

(R → C) ∧ (S → C)
is equivalent to (¬R ∨ C) ∧ (¬S ∨ C) (conditional law),
which is equivalent to (¬R ∧¬S) ∨ C (distributive law),
which is equivalent to ¬(R ∨ S) ∨ C (DeMorgan’s law),
which is equivalent to (R ∨ S) → C (conditional law).
You should read statements 1 and 4 again and see if it makes sense to you
that they’re equivalent.
5. ¬(R ∨ S) →¬C. This is the contrapositive of statement 3, so they are
equivalent. It is not equivalent to statements 1, 2, and 4.
Statements that mean P → Q come up very often in mathematics, but
sometimes they are not written in the form “If P then Q.” Here are a few other
ways of expressing the idea P → Q that are used often in mathematics:
P implies Q.
Q,if P.
P only if Q.
P is a sufficient condition for Q.
Q is a necessary condition for P.