Question

Write dual of the following Boolean Expression using Principle of Duality.
a) (X+Y)
b) XY + XY' + X'Y
c) A +A'B+B
d) (X+Y' + Z). (X+Y)

Answer 1

Dual of the given expressions are as follows:

• According to duality theorem the dual of a Boolean expression is obtained by interchanging the logical AND with logical OR and 0's with 1's (and vice-versa). For every Boolean function there exists a corresponding dual function.
• Hence the dual of each of the given expression turns out to be:

        a) (X+Y) = X.Y

        b) XY + XY' + X'Y= (X + Y) . (X + Y') . (X' + Y)

        c) A +A'B+B =  A . (A' + B) . B

        d) (X+Y' + Z). (X+Y) =  (X . Y' . Z) + (X . Y)

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Answer 2

What is Boolean Expression?

• A Boolean expression is a logical statement that is either TRUE or FALSE .
• Boolean expressions can compare data of any type as long as both parts of the expression have the same basic data type.
• You can test data to see if it is equal to, greater than, or less than other data.

What is Principle of Duality ?

• This principle states that any algebraic equality derived from these axioms will still be valid whenever the OR and AND operators, and identity elements 0 and 1, have been interchanged. i.e. changing every OR into AND and vice versa, and every 0 into 1 and vice versa.

THEN ACCORDING THE BOOLEAN EXPRESSION GIVEN

a)  XY’

(b)  (X + Y)(X + Y’)(X’ + Y)

(c)  A . (A’ + B) . B’

(d)  XY’Z + XY

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