I won't reproduce all the laws of boolean algebra here, my fingers got too tired. As luke was told, "use the force" meaning "google".

(some) theorems

X + 0 = x

x + 1 = 1

x + x = x

x * x = x

X * 1 = x

x * 0 = 0

(x')' = x

laws of complementarity

x + x' = 1

x * x' = 0

commutative, distr. assoc. laws

xy = yx

c + b = b + c

the distributive law not applicable to normal algebra is: x + yz = (x + y)(x + z). this is very useful in manipulating expressions.

de-morgan's laws

do not confuse with dual; in dual, change and to or, or to and, 0 to 1, 1 to 0 but do not complement variables. In contrast, to apply deMorgan's laws do what's done for a dual and also complement the variables.

DeMorgan's laws are :

(x + y)' = x'y'

(xy)' = x' + y'

In other words, (xy)' = x' + y' and the reverse (x + y)' = x'y' is also true.

example:

[(a' + b)c']' = (a' + b)' + c = ab' + c

another example:

(a'b + ab')' = (a'b)'(ab')' = (a + b')(a' + b) = aa' + ab + b'a' + b'b = ab + b'a'

two more examples:

(x + y)' = x'y'

[(ab' + c)d' + e]' = [(ab' +c)d']'e'

[(ab' + c)' + d]e' = [(ab')'c' +d]e'

(xy)' = x' + y'

[(ab' +c)d']'e' = [(ab' + c)' + d]e'

[(ab')'c' +d]e' = [(a' + b)c' +d]e'