What Is Boolean Algebra?

A British Mathematician George Boole introduced algebra of binary numbers in 1854. This is known as Boolean algebra. Boolean algebra is used in designing of logical circuits inside the computer. These circuits perform different types of logical operations. Thus Boolean algebra is also known as logical algebra or switching algebra.

The mathematical expressions of Boolean algebra are called Boolean expression. Boolean algebra describes the Boolean expression used in logical circuits. The Boolean expressions are simplified by means of basic theorems. The expressions that describe the logical circuits are also simplified by using Boolean algebra.

Boolean algebra differs from ordinary algebra in the different ways. Boolean algebra deals with binary numbers (0 and 1) while ordinary algebra deals with real numbers. Boolean algebra has only two basic operations cross, dot and complement operator. There are no subtraction and division. Ordinary algebra performs all arithmetic operations such as plus, minus, multiplication, division, subtraction etc. ordinary algebra does not have complement operation. Distributed law of Boolean algebra (a + (b . c) = (a + b). (a + c)) does not hold good in ordinary algebra. Boolean algebra perform only two basic operations which are called Boolean operators.

George Boole (1815 - 1864) was an English school teacher, little known in his days. To amuse himself, Boole invented what he called algebra of concepts – a way of writing down concepts in a formal language, and then solving them as one would solve an algebraic equation. For Boole, concepts could be viewed as sets, groups of ideas or objects. A set is a group of objects or concepts that share one or more common elements. For example, among flowers, red flowers are a set.

Boole identified three ways to identified three ways to describe the contents of a set. Speaking of the things that one might find in the garden, one can identify the following.

• Red flowers. In this set, one finds only those flowers that are red. The red garden trowel is not included because it is not a flower (Boolean operator AND)
• Red Objects or flowers. This set includes the red trowel as well as the red flowers. Anything that's red gets included (Boolean operator OR)
• Flowers, not red. This set includes flower of any colour, as long as they are not red (Boolean operator NOT)
Boolean algebra holds the key to computer circuit design and today Boolean Algebra is used to design microprocessors, which solves problems just as Boole foresaw – by carrying out logical operations.

This is a simple example of the use of Boolean algebra. Two of the operands used are AND (denoted with any multiplication symbol) and OR (denoted with addition symbol '+'). The point of Boolean algebra is to calculate if a set of conditions will make something happen. There are two possible values for the variables. '1' is true and '0' is false.

For example, if you say, "I will go to the movies if my best friend goes, if I can mooch some money from mom, AND if my father forks over his Ford." We can represent this statement with Boolean algebra. Let B=best friend going, M=mom gives money, F=father lends Ford, and G=am I going? Notice that the statement is AND, meaning ALL of these conditions must be met. Therefore, the expression is:

G = B*M*F

If your friend goes, father gives car, but mom doesn't give money, B=1, M=0, F=1

G = 1*0*1 = 0 = Not going to movie (notice AND is like multiplication)

If you had said, "I will go to the movies if my best friend goes, if I can mooch some money from mom, OR if my father forks over his Ford," the expression is G = 1+0+1 = 1 = Going to movie (notice OR is unlike addition) If any condition is true, the whole expression is true.