| [Note for Mac users] |
| 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)) | ||