The boolean expression $$Z = \bar{A}\bar{B}\bar{C}D + \bar{A}BC\bar{D} + A\bar{B}\bar{C}D + AB\bar{C}\bar{D}$$​ can be minimized to?

Correct option is C

Given expression:

​$$Z = \bar{A}\bar{B}\bar{C}D + \bar{A}BC\bar{D} + A\bar{B}\bar{C}D + AB\bar{C}\bar{D}$$​​

Step 1: Combine similar terms

​$$\bar{A}\bar{B}\bar{C}D + A\bar{B}\bar{C}D$$​​

Factor common variables:

​$$= \bar{B}\bar{C}D(\bar{A}+A)$$​​

Using Boolean identity:

​$$A + \bar{A} = 1$$​​

Thus:

​$$\bar{B}\bar{C}D(\bar{A}+A) = \bar{B}\bar{C}D$$​​

Now the expression becomes:

​$$Z = \bar{B}\bar{C}D + \bar{A}BC\bar{D} + AB\bar{C}\bar{D}$$​​

Step 2: Factor common terms

​$$\bar{A}BC\bar{D} + AB\bar{C}\bar{D}$$​​

Factor $$B\bar{D}:$$​​

​$$= B\bar{D}(\bar{A}C + A\bar{C})$$​​

Substituting back:

​$$Z = \bar{B}\bar{C}D + B\bar{D}(\bar{A}C + A\bar{C})$$​​

This simplifies to the minimized form:

​$$Z = \bar{A}BC\bar{D} + \bar{B}D + B\bar{C}\bar{D}$$​​

Thus the correct minimized expression corresponds to option (c).

Important Key Points:

1. The Boolean identity $$A + \bar{A} = 1$$​ is commonly used in Boolean algebra simplification.
2. Common factorization helps reduce complex Boolean expressions.
3. Simplification reduces the number of logic gates required in digital circuits.
4. Terms differing in only one literal can often be combined to obtain a minimal expression.

Knowledge Booster:

• (a) Incorrect because the term $$\bar{A}BC$$​ cannot be derived from the original Boolean expression.
• (b) Incorrect because it represents a partially simplified expression, not the minimal one.
• (d) Incorrect because the term AB cannot be obtained through valid Boolean simplification.