Step 1: Understanding the Boolean expression

The given Boolean expression is:

$A'B'C'D' + A'BC'D + A'B'C D + A'B'C D' + A'BC$

Here, the terms represent the following:

• $A'$: A is 0 (false)
• $B'$: B is 0 (false)
• $C'$: C is 0 (false)
• $D'$: D is 0 (false)

Step 2: Filling in the Karnaugh map (K-map)

We will map the terms on a 4-variable K-map (for variables A, B, C, and D). The values for A' (A = 0) apply to the top two rows.

K-map layout for $A, B, C, D$:

AB/CD	00	01	11	10
00
01
11
10

Now, we assign the Boolean terms to the corresponding cells in the K-map.

1. $A'B'C'D'$ maps to row 00 and column 00.
2. $A'BC'D$ maps to row 01 and column 01.
3. $A'B'C D$ maps to row 00 and column 11.
4. $A'B'C D'$ maps to row 00 and column 10.
5. $A'BC$ maps to row 01, and we cover both columns 10 and 11 (because $D$ can be 0 or 1).

Filled K-map:

AB/CD	00	01	11	10
00	1	0	1	1
01	0	1	1	1
11	0	0	0	0
10	0	0	0	0

Step 3: Simplification

We look for groups of 1s (power of 2: single cells, pairs, quads, or octets) and simplify:

1. Quad of 1s: The four 1s in the bottom-left (first two columns in rows 00 and 01) form a group. This corresponds to the simplified term $A'C'$.
2. Pair of 1s: The pair in row 01 and columns 10 and 11 gives $A'B$.

Final Simplified Expression:

$A'C' + A'B$

Conclusion

The simplified Boolean expression is $A'C' + A'B$.

Would you like more details or explanations on any of the steps?

Related Questions:

1. What are the advantages of using a Karnaugh map for simplification?
2. Can you explain the grouping rules in a K-map?
3. How do you handle don’t care conditions in a Karnaugh map?
4. How does the K-map change for expressions with more than four variables?
5. What is the difference between a Karnaugh map and Quine-McCluskey method for Boolean simplification?

Tip: Always look for the largest possible groups in a K-map to achieve the maximum simplification.