Up: Math 99 Selected Course Notes

Summary of What to Prove:

 To Prove: Do: A Í B Prove ("x)(xÎA Þ 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). ("xÎX)(P(x)) "Let x be an arbitrary ..." Prove P(x). (\$xÎX)(P(x)) "Take x = ..." Prove P(x) for this x. P(x) Þ 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) Û Q(x) Prove (P(x) Þ Q(x)) Ù (Q(x) Þ P(x)), or prove (P(x) Þ Q(x)) Ù (~P(x) Þ ~Q(x)), or prove (~Q(x) Þ ~P(x)) Ù (Q(x) Þ P(x)), or prove (~Q(x) Þ ~P(x)) Ù (~P(x) Þ ~Q(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). ~("xÎX)(P(x)) Prove (\$xÎX)(~P(x)). ~(\$xÎX)(P(x)) Prove ("xÎX)(~P(x)). ~(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))

Up: Math 99 Selected Course Notes