Boolean logic truth table

Given the variables p and q being both either true (1) or false (0), there are 16 possible outcomes combining them with an expression.

Expression	p=1,q=1	p=1,q=0	p=0,q=1	p=0,q=0
1	1	1	1	1
p or q	1	1	1	0
p or not q	1	1	0	1
p	1	1	0	0
not p or q	1	0	1	1
q	1	0	1	0
not (p xor q)	1	0	0	1
p and q	1	0	0	0
not p or not q	0	1	1	1
p xor q	0	1	1	0
not q	0	1	0	1
not p and q	0	1	0	0
not p	0	0	1	1
p and not q	0	0	1	0
not p and not q	0	0	0	1
0	0	0	0	0

Note that (not p or q) is equivalent to the expression (if p then q) or (p <= q)
(not(p xor q)) is equivalent to (if and only if p then q) or (p=q)
(p and not q) is equivalent to (p > q)
(p xor q) is equivalent to (p != q)

For expressions with more than two variables, the problem becomes soon complex. The problem, wether an expression with n boolean variables can be true with any combination of these variables hard to solve (see Boolean satisfiability problem).