To simplify the given Boolean functions using a Karnaugh Map (K-map), we follow these steps:

Function (a) $f_1 = \sum(1, 2, 5, 6)$

We will map the minterms to a 3-variable K-map and simplify.

1. Variables: Let the three variables be $A$, $B$, and $C$.
2. K-map setup: The K-map for three variables consists of 8 cells representing all possible combinations of $A$, $B$, and $C$.
   • Rows represent combinations of $A$ and $B$, while columns represent $C$.
   • The minterms are placed into their respective cells.

AB \ C	0	1
00	0	1
01	2	3
11	6	7
10	4	5

• Minterms: $f_1 = \sum(1, 2, 5, 6)$

AB \ C	0	1
00	0	1
01	1	0
11	1	0
10	0	1

Grouping:

• Group 1: Minterms $1, 5$ (form a vertical pair).
• Group 2: Minterms $2, 6$ (form another vertical pair).

Simplified Expression:

• Group $1, 5$: $\overline{A} C$
• Group $2, 6$: $B \overline{C}$

Thus, the simplified expression for $f_1$ is: $f_1 = \overline{A}C + B\overline{C}$

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Function (b) $f_2 = \sum(0, 1, 2, 3, 7)$

1. Variables: Again, let the three variables be $A$, $B$, and $C$.
2. K-map setup: Same as before.

AB \ C	0	1
00	0	1
01	2	3
11	6	7
10	4	5

• Minterms: $f_2 = \sum(0, 1, 2, 3, 7)$

AB \ C	0	1
00	1	1
01	1	1
11	0	1
10	0	0

Grouping:

• Group 1: Minterms $0, 1, 2, 3$ (form a 2x2 block).
• Group 2: Minterm $7$ (stands alone).

Simplified Expression:

• Group $0, 1, 2, 3$: $\overline{A}$
• Minterm $7$: $AB C$

Thus, the simplified expression for $f_2$ is: $f_2 = \overline{A} + ABC$

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Final Simplified Expressions:

• $f_1 = \overline{A}C + B\overline{C}$
• $f_2 = \overline{A} + ABC$

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Let me know if you'd like further details or clarifications!

Here are some related questions you might explore:

1. How do we derive Boolean expressions from a K-map with more than three variables?
2. How would the K-map look for a four-variable Boolean function?
3. What are the grouping rules when simplifying K-maps?
4. Can Karnaugh maps handle functions with "don't care" conditions?
5. What is the difference between minterms and maxterms in Boolean algebra?

Tip: Always check for the largest possible groupings in a K-map to achieve the most simplified expression.