1. f(A, B, C, D): Prime Implicants: A'B'D, A'CD, BCD, ABCD (underline A'CD for 11)
Minimum SOP: A'CD + BCD
2. f(A, B, C, D): Prime Implicants: A'B'D, A'BD, AB'D, ABCD (underline A'BD for 2 and AB'D for 13)
Minimum SOP: A'BD + AB'D
3. f(A, B, C, D): Prime Implicants: CD, A'B'D, A'CD, BCD, ABCD (underline CD for 4)
Minimum SOP: CD + A'B'D
4. f(A, B, C, D): Prime Implicants: A'B, AB'D, BCD, ABCD (underline A'B for 0, 1, 2)
Minimum SOP: A'B + AB'D
Prime Implicants and Minimum Sum-of-Products Expressions:
1. f(A, B, C, D) = Σ m(4, 11, 12, 13, 14) + Σ d(5, 6, 7, 8, 9, 10)
Prime Implicants: A'B'CD, ACD, ABCD, BC'D, BD, B'C'D.
Minimum Sum-of-Products Expressions:
1) Essential PI underlined: ACD + BD + B'C'D (minterms 4, 14 are covered only by ACD, minterm 13 only by BD, and minterm 6 only by B'C'D).
2) Other Equivalent Expressions: AB'CD + ACD + BC'D (using B'C'D instead of BD + B'C'D, covering the same minterms with fewer literals).
2. f(A, B, C, D) = Σ m(1, 2, 4, 13, 14) + Σ d(5, 6, 7, 8, 9, 10)
Prime Implicants: ABC'D, AB'CD, ACD, ABCD, BC'D, BD.
Minimum Sum-of-Products Expressions:
1) Essential PI underlined: ABC'D + ACD + BD (minterm 1 is covered only by ABC'D, minterm 13 only by ACD, and minterm 2 only by BD).
2) Other Equivalent Expressions: A'BC'D + ACD + BD (using A'BC'D instead of AB'CD + BD, covering the same minterms with fewer literals).
3. f(A, B, C, D) = Σ m(4) + Σ d(5, 6, 7, 8, 9, 10, 11, 12, 13, 14)
Prime Implicants: ACD, ABCD, BC'D, BD, B'C'D.
Minimum Sum-of-Products Expressions:
1) Any combination of 3 out of 5 prime implicants will suffice, as they cover all minterms. No essential PIs.
Examples: ACD + BD + B'C'D, BC'D + BD + ABCD, etc.
4. f(A, B, C, D) = Σ m(4, 15) + Σ d(0, 1, 2, 5, 6, 7, 8, 9, 10)
Prime Implicants: A'BCD, AB'CD, ABCD, BC'D, BD, B'C'D.
Minimum Sum-of-Products Expressions:
1) Essential PI underlined: ABCD + B'C'D (minterm 15 is covered only by ABCD, and minterm 0 only by B'C'D).
2) Other Equivalent Expressions: A'BC'D + ABCD + BD (using A'BC'D instead of AB'CD + BD, covering the same minterms with fewer literals).