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Up: Math 99 Selected Course Notes

Summary of What to Prove:

To Prove:Do:
A B Prove ("x)(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).
("xX)(P(x)) "Let x be an arbitrary ..."
Prove P(x).
($xX)(P(x)) "Take x = ..."
Prove P(x) for this x.
P(x) Q(x) "Assume P(x) is true," prove Q(x) is true, or
"Assume Q(x) is false," prove P(x) is false, or
"Assume P(x) is true and Q(x) is false", produce a contradiction.
P(x) Q(x) Prove (P(x) Q(x)) (Q(x) P(x)), or
prove (P(x) Q(x)) (~P(x) ~Q(x)), or
prove (~Q(x) ~P(x)) (Q(x) P(x)), or
prove (~Q(x) ~P(x)) (~P(x) ~Q(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).
~("xX)(P(x)) Prove ($xX)(~P(x)).
~($xX)(P(x)) Prove ("xX)(~P(x)).
~(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))


Up: Math 99 Selected Course Notes

Comments to: dpvc@union.edu
Created: 28 Apr 1999 --- Last modified: Aug 27, 1999 10:23:27 AM