Math 99 (Notes)

# Summary of What to Prove:

 To Prove: Do: P Þ 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 Û Q Prove (P Þ Q) Ù (Q Þ P), or prove (P Þ Q) Ù (~P Þ ~Q), or prove (~Q Þ ~P) Ù (Q Þ P), or prove (~Q Þ ~P) Ù (~P Þ ~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) Ù (B Í A). A = Ø Prove ("x)(xÏA) (frequently best to use proof by contradiction). xÎAÈB Prove (xÎA) Ú (xÎB). xÎA Ç B Prove (xÎA) Ù (xÎB). xÎA - B Prove (xÎA) Ù (xÏB). ~(P(x) Þ Q(x)) Prove (\$x)(P(x) Ù ~Q(x)). ~(P(x) Û Q(x)) Prove ("x)(P(x) Ù ~Q(x)) Ú (\$x)(Q(x) Ù ~P(x)) ~(\$x)(P(x)) Prove ("x)(~P(x)). ~("x)(P(x)) Prove (\$x)(~P(x)). AËB Prove (\$x)(xÎA Ù xÏB). A ¹ B Prove (A Ë B) Ú (B Ë A). ie, there is an xÎA where xÏB or there is an xÎB where xÏA. A ¹ Ø Prove (\$x)(xÎA). xÏA È B Prove (xÏA) Ù (xÏB). xÏA Ç B Prove (xÏA) Ú (xÏB). xÏA - B Prove (xÏA) Ú (xÎB).

 Math 99 (Fall 2001) web pages Created: 28 Apr 1999 Last modified: Oct 4, 2001 9:41:04 PM Comments to: `dpvc@union.edu`