Question

Consider the following Boolean expression. \[ F = (X + Y + Z)(\overline{X} + Y)(\overline{Y} + Z) \] Which of the following Boolean expressions is/are equivalent to $\overline{F$ (complement of $F$)?}

• A. $(\overline{X} + \overline{Y} + \overline{Z})(X + \overline{Y})(Y + \overline{Z})$
• B. $X\overline{Y} + \overline{Z}$
• C. $(X + \overline{Z})(\overline{Y} + \overline{Z})$
• D. $X\overline{Y} + Y\overline{Z} + \overline{X}\,\overline{Y}\,\overline{Z}$

Hint:

To find equivalent Boolean expressions, first compute the complement using De Morgan's laws and then simplify using absorption and distributive properties.

Answer 1

The Correct Option is B, C, D

Step 1: Find the complement of the function.
Given:

F = (X + Y + Z)(X + Y)(Y + Z)

Taking complement and applying De Morgan’s theorem:

F = (X + Y + Z) + (X + Y) + (Y + Z)

= (X Y Z) + (XY) + (YZ)

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Step 2: Match with the given options.

Option (B): XY + Z
This expression is obtained from the above form using absorption laws, hence it is equivalent.

Option (C): (X + Z) (Y + Z)
On expansion, this reduces to the same sum-of-products form as F, so it is equivalent.

Option (D): XY + YZ + X Y Z
This matches exactly with the derived expression for F.

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Step 3: Reject the incorrect option.
Option (A): Does not simplify to the obtained Boolean expression and is therefore not equivalent.

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Final Conclusion:
The Boolean expressions equivalent to F are (B), (C), and (D).