Sample Midterm #1 Solution
1.
Prove using Boolean algebra that a′c′ + ab + ac + a′b′ = a′c′ + ab + b′c. Write the
particular law you are using in each step.
Proof:
a′c′ + ab + ac + a′b′
= a’c’ + ab + (ac + a’b’) associativity
= a’c’ + ab + (ac + a’b’ + b’c) consensus
= a’c’ + ab + ac + a’b’ + b’c + b’c idenpotency
= (a’c’ + b’c + a’b’) + (ab + b’c + ac) commutativity+ associativity
= (a’c’ + b’c) + (ab + b’c) consensus
= a’c + ab + b’c idenpotency
2. Prove using Boolean algebra that (a + c)(a′ + c′)(b′ + c + d′)(a + b′ + d′) = (a
+ c)(a′ + c′)(b′ + d′). Write the particular law you are using in each step.
Proof: (a + c)(a′ + c′)(b′ + c + d′)(a + b′ + d′)
= (a + c) (a’ + c’) ((ac) + (b’ + d’)) distributivity
= (a + c) (a’ + c’) ac + (a + c) (a’ + c’) (b’ + d’) distributivity
= (a + c) (a’ac + c’ac) + (a + c) (a’ + c’) (b’ + d’) distributivity
= (a + c) (0 + 0) + (a + c) (a’ + c’) (b’ + d’) complement
= 0 + (a + c) (a’ + c’) (b’ + d’) nullity
= (a + c) (a’ + c’) (b’ + d’) identity
3. Use Karnaugh map to simplify function f (a, b, c, d) =
Σ
m(0,1, 2, 3, 4, 5, 7,8,12)
+
Σ
d(10,11). List all possible minimal two-level sum of products expressions.
Show the switching functions. No need for the diagram.
F = a’d + a’b’ + c’d’
or F = a’d + b’c + c’d’
or F = a'd + c'd' + b'd'
1
1
1
1
1
1
1
1
x
1
x
00
01
11
10
00
01
11
10
cd
ab
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