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Boolean Expression

Consider a Boolean algebra (B, ∨,∧,',0,1).A Boolean expression over Boolean algebra B is defined as

  1. Every element of B is a Boolean expression.
  2. Every variable name is a Boolean expression.
  3. If a1 and a2 are Boolean expression, then a1,'∨ a2 and a1∧ a2 are Boolean expressions.

Example: Consider a Boolean algebra ({0, 1, 2, 3},∨,∧,',0,1).

  1. 0∨x
  2. (2∧3)
  3. (x1∨ x2)∧ (x1∧x3)

are Boolean expressions over the Boolean Algebra.

    A Boolean expression that contains n distinct variables is usually referred to as a Boolean expression of n variables.

Evaluation of Boolean Expression:

Let E (x1,x2,....xn)be a Boolean Expression of n variables over a Boolean algebra B. By an assignment of values to the variables x1,x2,....xn, means an assignment of elements of A to be the values of the variables.

    We can evaluate the expression E ( x1,x2,....xn) by substituting the variables in the expression by their values.

Example: Consider the Boolean Expression

            E( x1,x2,x3)=(x1∨ x2 )∧(x1∨ x2 )∧(x2∨ x3)

over the Boolean algebra ({0,1}, ∨,∧,')

By assigning the values x1=0,x2=1,x3=0 yields

            E (0, 1, 0) = (0∨1)∧(0∨1)∧(1∨0)=1∧1∧0=0.

Equivalent Boolean Expressions:

Two Boolean expressions of n variables are said to be equal if they assume the same value for every assignment of values to the n variables.

Example: The following two Boolean algebras (x1∧x2)∨(x1∧ x3 ) and x1∧ (x2∨ x3) are equivalent.

We may write E1( x1,x2,....xn)=E2( x1,x2,....xn) to mean the two expressions E1( x1,x2,....xn) and E2( x1,x2,....xn)are equivalent.

Example: The Boolean expression (x1∧x2∧x3)∨(x1∧x2 )∨(x1∧x3) over the Boolean algebra ({0, 1}, ∨,∧,') defines the function f in figure.

Min-term: A Boolean Expression of n variables x1,x2,....xn is said to be a min-term if it is of the form x1∧x2∧x3∧....∧xn

where xi is used to denote xi or xi'.

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