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CSE140 Midterm 1, October 14, 2010, Name___________________________
PID 
(I) (15pts) (Problem Formulation and Canonical Expression) A full adder inputs
three bits (
a
,
b
,
cin
) and outputs the sum bit and carry bit (
s
,
cout
). Write the
truth table of the full adder and the corresponding canonical
productof
maxterms
expressions.
A
B
Cin
S
Cout
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1
S = (a + b + cin)(a + b' + cin')(a' + b + cin')(a' + b' + cin)
Cout = (a + b + c)(a' + b + cin)(a + b' + cin)(a + b + cin')
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(II) (10pts) (Laws and Theorems of Boolean Algebra) Specify the axioms or
theorems for each step. Prove using Boolean algebra that
a
’
b
+
a’c’
+
b’c’
=
a’b
+
b’c’
.
a’b + a’c’ +b’c’
= a’b + a’c’(b+b’) + b’c’
(complementarity)
= a’b + a’c’b + a’c’b’ + b’c’
(distributivity)
= a’b + b’c’ (Absorptio
n of the first two and last two terms)
Can also use Consensus directly to prove the result.
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