To express the function $f(a, b, c, d)$ as a sum of minterms, where $f(a, b, c, d) = \sum(0, 1, 3, 4, 5, 7, 9, 11, 15)$, we will use the minterm representation.

Each number in the sum notation represents a minterm where the binary form of the number indicates the values of $a$, $b$, $c$, and $d$ respectively. Here’s the breakdown:

Step 1: Convert the minterm indices to binary

Let's convert each of the indices (0, 1, 3, 4, 5, 7, 9, 11, and 15) to binary to find the corresponding minterms:

1. $0$ in binary: $0000$ ( $a'b'c'd'$ )
2. $1$ in binary: $0001$ ( $a'b'c'd$ )
3. $3$ in binary: $0011$ ( $a'b'cd$ )
4. $4$ in binary: $0100$ ( $a'bc'd'$ )
5. $5$ in binary: $0101$ ( $a'bc'd$ )
6. $7$ in binary: $0111$ ( $a'bcd$ )
7. $9$ in binary: $1001$ ( $ab'c'd$ )
8. $11$ in binary: $1011$ ( $ab'cd$ )
9. $15$ in binary: $1111$ ( $abcd$ )

Step 2: Write the function as a sum of minterms

Now we can write $f(a, b, c, d)$ in terms of these minterms:

$f(a, b, c, d) = a'b'c'd' + a'b'c'd + a'b'cd + a'bc'd' + a'bc'd + a'bcd + ab'c'd + ab'cd + abcd$

Step 3: Simplify (if needed)

If simplification is needed, we can apply Boolean algebra rules or use a Karnaugh map for a more compact expression.

Would you like a detailed explanation on simplification, or is there anything specific you'd like help with?

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Additional Questions to Explore:

1. How can Karnaugh maps be used to simplify Boolean expressions?
2. What is the difference between sum of minterms and product of maxterms?
3. How can we convert Boolean functions from canonical form to simplified form?
4. What methods exist for simplifying Boolean expressions (e.g., Karnaugh maps vs. Boolean algebra)?
5. How does this function behave when some variables are fixed to certain values?

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Tip: Using binary representation of minterm indices is a quick way to understand the variable assignments in Boolean functions.