Math 99 (Notes)
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Summary of What to Prove:

To Prove:Do:
P Q "Assume P is true," prove Q is true, or
"Assume Q is false," prove P is false, or
"Assume P is true and Q is false", produce a contradiction.
P Q Prove (P Q) (Q P), or
prove (P Q) (~P ~Q), or
prove (~Q ~P) (Q P), or
prove (~Q ~P) (~P ~Q)
("x)(P(x)) "Let x be an arbitrary ..."
Prove P(x).
($x)(P(x)) "Take x = ..."
Prove P(x) for this x.
A B Prove ("xA)(xB)
i.e., if xA then xB.
A = B Prove (A B) (B A).
A = Ø Prove ("x)(xA)
(frequently best to use proof by contradiction).
xAB Prove (xA) (xB).
xA B Prove (xA) (xB).
xA - B Prove (xA) (xB).

~(P(x) Q(x)) Prove ($x)(P(x) ~Q(x)).
~(P(x) Q(x)) Prove ("x)(P(x) ~Q(x)) ($x)(Q(x) ~P(x))
~($x)(P(x)) Prove ("x)(~P(x)).
~("x)(P(x)) Prove ($x)(~P(x)).
AB Prove ($x)(xA xB).
A B Prove (A B) (B A).
ie, there is an xA where xB or there is an xB where xA.
A Ø Prove ($x)(xA).
xA B Prove (xA) (xB).
xA B Prove (xA) (xB).
xA - B Prove (xA) (xB).

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Created: 28 Apr 1999
Last modified: 05 Oct 2000 12:34:44
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