COT3100C-01, Fall 2002
S. Lang
Solution Keys to Assignment #2 (40 pts.)
9/20/2002
1.
(10 pts., 2 pts. each part) The following Venn diagram shows three sets
A
(red)
,
B
(blue)
, and
C
(green)
contained within a universe set named
U
(violet)
.
In the diagram, different subsets
(regions) are labeled by the numbers 1 through 8.
Thus, set
A
contains subsets labeled 1, 2, 4,
and 5; set
B
contains subsets labeled 2, 3, 5, and 6; set
C
contains subsets labeled 4, 5, 6, and 7;
set
U
contains subsets labeled 1 through 8 (i.e., the universe).
Now, give the labels (i.e.,
numbers) of all the subsets that are contained in each of the following sets:
2.
(24 pts., 4 pts. each part) Prove each of the following statements (a) – (f), assuming the symbols
A
,
B
, and
C
represent sets contained in a universe set
U
.
You are allowed to use appropriate
definitions, and the following theorems and laws (
but
only these
) in the proof.
Be sure to
explain each step of your proof
.
(Commutative Law)
A
∪
B =
B
∪
A
,
A
∩
B
=
B
∩
A
.
(Associative Law) (
A
∪
B
)
∪
C
=
A
∪
(
B
∪
C
),
(
A
∩
B
)
∩
C
=
A
∩
(
B
∩
C
).
(Distributive Law)
A
∪
(
B
∩
C
) = (
A
∪
B
)
∩
(
A
∪
C
),
A
∩
(
B
∪
C
) = (
A
∩
B
)
∪
(
A
∩
C
).
(Idempotent Law)
A
∪
A
=
A
,
A
∩
A
=
A
.
(Absorption Law)
A
∩
(
A
∪
B
) =
A
,
A
∪
(
A
∩
B
) =
A
. In particular,
A
∩
U
=
A
, and
A
∪
U
=
U
,
where
U
denotes the universe.
(De Morgan’s Law)