## Boolean algebra

**Boolean Algebra:**

** **A Boolean algebra is an algebra consisting of a set B with >= 2 elements, together with three operations. They are

- AND operation – Boolean product
- OR operation- Boolean sum
- NOT operation-Complement

Boolean algebra is an algebra that deals with the binary variables and logic operations. Defines on the set, such that for any element a, b, c, …… of set B, a.b, a+b, and a’ are in B. Consider the two element Boolean algebra B={(0,1); .,+,’;0,1}. The operations AND, OR and NOT are defined as follows.

Boolean algebra is a switching algebra that deals with the binary variables and logic operations. A Boolean function can be expressed algebraically with binary variables, the logic operation symbols, parenthesis, and equal sign. For a given values of variables, the Boolean function can be either 1 or 0.

Consider, for example, the Boolean function F=x+y’z. the function F is equal to 1 if x is 1 or both if y’ and z are equal to 1. F is equal to 0 otherwise. But saying that y’=1 is equivalent to saying that y=0 since y’ is the complement of y. Therefore, we may say that F=1 if x=1 or if yz=01. The relationship between a function and its binary variables can be represented in a truth table.

A Boolean function can be transformed from an algebraic expression into a logic diagram composed of AND, OR, and inverter gates. In a logic diagram, variables of the function are taken to be the inputs of the circuit, and the variable symbol of the function is taken as the output of the circuit. The purpose of Boolean algebra is to facilitate the analysis and design of digital circuits. It provides a convenient tool to:

- Express in algebraic form a truth table relationship between binary variables
- Express in algebraic form the input output relationship of logic diagrams
- Find simpler circuits for the same function

By manipulating the Boolean expression according to Boolean algebra rules, one may obtain a simpler expression that will require fewer gates. To see how this is done, we must first study the manipulative capabilities of Boolean algebra. All the identities in the table can be proven by means of truth tables.

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