Which among the following is the dual of Boolean expression X+YZ=(X+Y) (X+Z)? 

Explanation:

Dual of a Boolean Expression

Definition: The dual of a Boolean expression is derived by replacing all AND (×) operations with OR (+) operations, all OR (+) operations with AND (×) operations, and swapping the constants 1 and 0 in the original expression. The variables in the Boolean expression remain unchanged.

To determine the dual of a given Boolean expression, follow these steps:

1. Identify all the AND (×) and OR (+) operations in the expression.
2. Replace AND (×) with OR (+) and vice versa.
3. Swap 1 with 0 wherever they appear in the expression.

Given Boolean Expression:

X + YZ = (X + Y) × (X + Z)

Let’s determine the dual of this expression:

• In the given expression, the OR (+) operation between X and YZ will be replaced with an AND (×) operation.
• The AND (×) operations within YZ and (X + Y) × (X + Z) will be replaced with OR (+) operations.
• Since there are no constants 1 or 0 in the expression, we do not need to swap them.

Replacing operations step by step:

• The left-hand side, X + YZ, becomes X × (Y + Z).
• The right-hand side, (X + Y) × (X + Z), becomes (X × Y) + (X × Z).

Thus, the dual of the given Boolean expression is:

X × (Y + Z) = (X × Y) + (X × Z)

Correct Option Analysis:

The correct option is:

Option 1: X × (Y + Z) = XY + YZ

This is the correct dual of the given Boolean expression. As derived above, the left-hand side X + YZ becomes X × (Y + Z), and the right-hand side (X + Y) × (X + Z) becomes (X × Y) + (X × Z). Hence, the correct dual is accurately represented in Option 1.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: X.(Y + Z) = X.Y + X.Z

This option is not the correct dual of the given Boolean expression. While it might seem similar, it actually represents the distributive property of Boolean algebra and does not match the dual derived from the original expression.

Option 3: X + (Y + Z) = X.Y + Z

This option introduces an additional OR (+) operation within (Y + Z) that was not present in the original expression. Moreover, the right-hand side, X.Y + Z, does not correspond to the structure of the dual derived from the given Boolean expression. Therefore, it is incorrect.

Option 4: X + (YZ) = X + Y + Z

This option does not align with the dual derived from the given Boolean expression. It incorrectly simplifies X + (YZ) to X + Y + Z, which is not mathematically valid based on Boolean algebra rules. Hence, this option is incorrect.

Conclusion:

By understanding the concept of duality in Boolean algebra, we can accurately derive the dual of any given Boolean expression. The dual is obtained by swapping AND (×) and OR (+) operations and interchanging the constants 1 and 0. For the given expression X + YZ = (X + Y) × (X + Z), the correct dual is X × (Y + Z) = (X × Y) + (X × Z), which is represented in Option 1.