The following is an excerpt from Wikipedia's article titled Two-Element Boolean Algebra.

[In the expressions below, the multiplication operator looks like a period to make the typing easier. The multiplication operator is usually written midway up, just as in regular algebra.]

There are several names and notations for the two binary operations of Boolean algebra. Here they are called 'sum' and 'product', notated by '+' and '.', respectively. As for order of operations, '.' precedes '+', but brackets may override. Hence A.B + C is parsed as (A.B) + C not A.(B + C).

Interpreting one of 0 and 1 as True and the other as False yields classical bivalent logic in equational form. In this case, '+' is read as OR, '.' as AND, and complementation as NOT.

Some basic identities

• 1+1 = 1+0 = 0+1 = 1
• 0+0 = 0
• 0.0 = 0.1 = 1.0 = 0
• 1.1 = 1
• 
• 

Note that:

• '+' and .' work exactly as in numerical arithmetic, except that 1+1=1.  '+' and '.' are derived by analogy from numerical arithmetic; simply set any nonzero number to 1.
• Swapping 0 and 1, and '+' and '.' preserves truth.

• A.B.C = B.C.A.
• 
• 

• A.A = A
• A + A = A
• A + 0 = A
• A + 1 = 1
• A.0 = 0
• A.1 = A
• 
• 
• 

Each of '+' and '.' distributes over the other. That is, A.(B+C) = A.B + A.C and A+B.C = (A+B).(A+C). 

De Morgan's Theorem

De Morgan's theorem states that if you do the following, in the given order, to any Boolean function:

• Complement every variable;
• Swap + and . operators (taking care to add brackets to ensure the order of operations remains the same);
• Complement the result,

the result is logically equivalent to what you started with. Repeated application of De Morgan's theorem to parts of a function can be used to drive all complements down to the individual variables.