# Summary of What to Prove:

 To Prove: Do: P Ţ 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 Ű Q Prove (P Ţ Q) Ů (Q Ţ P), or prove (P Ţ Q) Ů (~P Ţ ~Q), or prove (~Q Ţ ~P) Ů (Q Ţ P), or prove (~Q Ţ ~P) Ů (~P Ţ ~Q) ("x)(P(x)) "Let x be an arbitrary ..." Prove P(x). (\$x)(P(x)) "Take x = ..." Prove P(x) for this x. A Í B Prove ("xÎA)(xÎB) i.e., if xÎA then xÎB. A = B Prove (A Í B) Ů (B Í A). A = Ø Prove ("x)(xĎA) (frequently best to use proof by contradiction). xÎAČB Prove (xÎA) Ú (xÎB). xÎA Ç B Prove (xÎA) Ů (xÎB). xÎA - B Prove (xÎA) Ů (xĎB). ~(P(x) Ţ Q(x)) Prove (\$x)(P(x) Ů ~Q(x)). ~(P(x) Ű Q(x)) Prove ("x)(P(x) Ů ~Q(x)) Ú (\$x)(Q(x) Ů ~P(x)) ~(\$x)(P(x)) Prove ("x)(~P(x)). ~("x)(P(x)) Prove (\$x)(~P(x)). AËB Prove (\$x)(xÎA Ů xĎB). A ą B Prove (A Ë B) Ú (B Ë A). ie, there is an xÎA where xĎB or there is an xÎB where xĎA. A ą Ø Prove (\$x)(xÎA). xĎA Č B Prove (xĎA) Ů (xĎB). xĎA Ç B Prove (xĎA) Ú (xĎB). xĎA - B Prove (xĎA) Ú (xÎB). Math 99 (Fall 2000) web pages Created: 28 Apr 1999 Last modified: 05 Oct 2000 12:34:44 Comments to: `dpvc@union.edu`