Disproof by counterexample
Method of generalizing from the generic particular (exhaustion)
Method of Direct Proof:
1.
Express statement to be proved in the form “Ax E D, if P(x) then Q(x).” (often done mentally)
2.
Start proof by supposing x is arbitrary but particular element of D for which the hypothesis
P(x) is true. (This step is often abbreviated “Suppose x E D and P(x).”)
3.
Show that the conclusion Q(x) is true by using definitions, previously established results and
rules of logical inference
D  n =>
n=dk (d divides n)
N is a multiple of d, d is a factor of n, d is a divisor of n
QTR:
n=dq+r and 0 <= r < d where n and d are integers and q and r are unique integers
Method of proof by contradiction:
1.
Suppose the statement to be proved is false.
That is, suppose that the negation of the
statement is true
2.
Show that this supposition leads logically to a contradiction
3.
Conclude that the statement to be proved is true
Method of Proof by contraposition
1.
Express the statement to be proved in the form Ax in D, if P(x) then Q(x) (mentally)
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 Fall '08
 MITRA
 Logic, Integers, Mathematical Induction, Mathematical logic

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