MATH 74 QUIZ 1
February 6th, 2009
Name:
Show all work and attempt to write neatly. There is a total of 60 points.
1. (10 points) Let
A,B
be subsets of
U
. Prove that
A
S
B
= (
A
T
B
c
)
S
B
.
We have
(
A
±
B
c
)
[
B
=
{
x
∈
U

(
x
∈
A
∧
x
∈
B
c
)
∨
x
∈
B
}
=
{
x
∈
U

(
x
∈
A
∧
x /
∈
B
)
∨
x
∈
B
}
=
{
x
∈
U

(
x
∈
A
∨
x
∈
B
)
∧
(
x
∈
B
∨
x /
∈
B
)
}
where the second line follows from the distributive property of
∧
and
∨
. But
x
∈
B
∨
x /
∈
B
is
equivalent to
x
∈
U
, which is always true. Thus we ﬁnd
(
A
±
B
c
)
[
B
=
{
x
∈
U

(
x
∈
A
∨
x
∈
B
)
∧
(
x
∈
U
}
=
{
x
∈
U

x
∈
A
∨
x
∈
B
}
=
A
[
B.
2. (10 points) Let
P,Q,R
be propositions. Let
A
be the proposition
P
=
⇒
((
Q
=
⇒
R
) and (
R
=
⇒
Q
))
.
Let
B
be the proposition
(not
P
and (
Q
or
R
)) or (
P
and (
Q
and
R
))
.
Use truth tables to decide whether or not
A
and
B
are logically equivalent.
If
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 Spring '09
 DANBERWICK
 Logic, Division, Sets, X1

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