Boolean Expression:Consider a Boolean algebra (B, ∨,∧,',0,1).A Boolean expression over Boolean algebra B is defined as
Example: Consider a Boolean algebra ({0, 1, 2, 3},∨,∧,',0,1).
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 (x_{1},x_{2},....x_{n})be a Boolean Expression of n variables over a Boolean algebra B. By an assignment of values to the variables x_{1},x_{2},....x_{n}, means an assignment of elements of A to be the values of the variables. We can evaluate the expression E ( x_{1},x_{2},....x_{n}) by substituting the variables in the expression by their values. Example: Consider the Boolean Expression E( x_{1},x_{2},x_{3})=(x_{1}∨ x_{2} )∧(x_{1}∨ x_{2} )∧(x_{2}∨ x_{3}) over the Boolean algebra ({0,1}, ∨,∧,') By assigning the values x_{1}=0,x_{2}=1,x_{3}=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 (x_{1}∧x_{2})∨(x_{1}∧ x_{3} ) and x_{1}∧ (x_{2}∨ x_{3}) are equivalent. We may write E_{1}( x_{1},x_{2},....x_{n})=E_{2}( x_{1},x_{2},....x_{n}) to mean the two expressions E_{1}( x_{1},x_{2},....x_{n}) and E_{2}( x_{1},x_{2},....x_{n})are equivalent. Example: The Boolean expression (x_{1}∧x_{2}∧x_{3})∨(x_{1}∧x_{2} )∨(x_{1}∧x_{3}) over the Boolean algebra ({0, 1}, ∨,∧,') defines the function f in figure. Minterm: A Boolean Expression of n variables x_{1},x_{2},....x_{n} is said to be a minterm if it is of the form x_{1}∧x_{2}∧x_{3}∧....∧x_{n} where x_{i} is used to denote x_{i} or x_{i}'.
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