epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Introduction and Basic Terminology 11
for an animal to have four legs and not be a cow. On the other hand, being a
cow is a sufficient condition for knowing that the animal has four legs.
Consult the James & James Mathematics Dictionary if you want to find out
more about "sufficient" and "necessary" conditions.
All arguments having this form (called modus ponens) are valid. The
expression "modus ponens" comes from the Latin ponere, meaning "to
affirm."
Very often one of the so-called truth tables* is used to remember the
information just seen (T = true, F = false):
If A, then B
T T T
T F F
F T T
F F T
Because in a statement of the form "If A, then B" the hypothesis and
the conclusion are clearly separated (part A, the hypothesis, contains all the
information we are allowed to use; part B is the conclusion we want to reach,
given the previous information), it is useful to try to write in this form any
statement to be proved. The following steps can make the statement of a
theorem simpler and therefore more manageable, without changing its
meaning:
1. Identify the hypothesis (A) and conclusion (B) so the statement can be
written in the form "If A, then B" or "A impHes B."
2. Watch out for irrelevant details.
3. Rewrite the statement to be proved in a form you are comfortable with,
even if it is not the most elegant.
4. Check all relevant properties (from what you are supposed to know)
of the objects involved.
If you get stuck while constructing the proof, double-check whether you
have overlooked some explicit or implicit information you are supposed to
know and be able to use in the given context. As mentioned in the General
* Truth tables are diagrams used to analyze composite statements. A column is assigned to each
of the simple statements that form the composite statement, then one considers all possible
combinations of "true" and "false" for them. The logic connectives used to construct the
composite statement (e.g., and, or, if... then ...) will determine the truth value of the composite
statement.