Simplifying Boolean Algebra

Simplifying Boolean algebra is surprisingly easy when you understand Boolean logic. Boolean algebra is based on simple rules of logic, allowing equations to be linked together and ordered in specific ways. The basic rules are quite simple, but applying them can be a challenge unless you know what question a Boolean equation is asking.

• Every Boolean law has two expressions. "Duality" is the concept that every Boolean law has two expressions. You can get both expressions from Boolean law by changing AND to OR,  OR to AND, 1s to 0s and 0s to 1s. One expression may help where another doesn't, so consider both expressions of Boolean laws when simplifying.
• Use a combination of the various axioms to simplify Boolean algebra. You often need to use a combination of the various axioms to simplify a Boolean problem. Consider using combinations of laws to break complex expressions down into basic equations.
• Some Boolean algebra is similar to regular algebra. Some Boolean algebra rules are similar to regular algebra rules, but in some ways Boolean algebra is vastly different. If you understand the similarities and differences, you can utilize those similarities to help you simplify Boolean algebra. Be careful not to over-assume, though; Boolean algebra has different rules for solving problems, and applying normal algebra rules can get you into trouble.
• Get your operations right. Boolean algebra has different operations than regular algebra, so know which operations you're working with before you start simplifying. Different operations determine which laws you can use; you can't begin to simplify or move things around correctly unless you're using the proper operations. Know whether an equation calls for AND, OR, NAND, NOR, XOR or NOT, and which law to use with each operation.
• Boolean logic is used in electronics. When you're simplifying Boolean algebra, keep in mind that Boolean logic is used in electronics and in the design of computer circuits. If you get stumped, think about the way electronics work; they take electricity and move it through a series of wires and resistors, in much the same way that water moves through pipes. Without a complete circuit, the water doesn't flow. If there's a hole in the circuit, the water leaks out and doesn't go where its needed. Sometimes just knowing your desired result can help you select the right method, so keep your end-product in mind when simplifying Boolean problems.

The Five Laws of Boolean Algebra

Commutative Law
(a) A + B = B + A
(b) A B = B A

Associative Law
(a) (A + B) + C = A + (B + C)
(b) (A B) C = A (B C)

Distributive Law
(a) A (B + C) = A B + A C
(b) A + (B C) = (A + B) (A + C)

Identity Law
(a) A + A = A
(b) A A = A

Redundance Law
(a) A + A B = A
(b) A (A + B) = A