COT 310001
Feb. 14’2001
Test#1 (100 pts)
Solutions
Name:_________________
SSN:__________________
1. For each of the Venn diagrams write the set represented by shaded area in terms of set
operations on sets
A
,
B
and
C
. Try to simplify your answer to as few symbols as you can.
a) (4pts)
Ans:
A
∩
B

C
b) (4pts).
Ans:
B
∪
C

A
∩
C
c) (4pts)
Ans. (
B
∪
C
)

(
B
∩
C
)
1
A
B
C
A
B
C
A
B
C
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View Full Document2. (8pts). Use truth table method to determine whether the following statement is a
tautology, contradiction, or neither:
[
p
∧
(
q
∨¬
r
)]
∨
(
¬
p
∨
r
)
p
q
r
¬
p
¬
r
q
∨¬
r
p
∧
(
q
∨¬
r
)
¬
p
∨
r
[
p
∧
(
q
∨¬
r
)]
∨
(
¬
p
∨
r
)
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
0
1
1
0
1
0
0
0
0
1
1
1
0
0
0
1
1
1
0
1
0
1
1
1
0
1
0
1
1
0
1
0
1
1
1
0
1
1
0
0
1
1
0
0
0
1
1
0
0
0
1
1
1
0
1
1
(where 1 stands for True and 0 for False)
Since we have True in all rows of the last column, the statement [
p
∧
(
q
∨¬
r
)]
∨
(
¬
p
∨
r
)
is a tautology.
3. (10pts). Use laws of logic to prove that (
p
→
r
)
∧
(
q
→
r
) is equivalent to (
p
∨
q
)
→
r
.
Use one law at a step (except commutative and associative laws). Specify exactly what
you do at each step.
(
p
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 Spring '09
 Set Theory, Numerical digit, Natural number, A− B−, p∧ q∨¬

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