
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). 

