COT3100C01, Spring 2000
S. Lang
Solution Key to Assignment #2 (40 pts.)
Feb. 11, 2000
1.
(12 pts.) Prove each of the following statements in Parts (a)  (f), assuming the
symbols
A
,
B
, and
C
represent (arbitrary) sets.
Your proof must be based on the
appropriate definitions and the laws and theorems listed below (
but only these
).
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 Property)
A
∪
A
=
A
,
A
∩
A
=
A
.
(De Morgan’s Law)
¬
(
A
∪
B
) =
¬
A
∩
¬
B
,
¬
(
A
∩
B
) =
¬
A
∪
¬
B
.
(Double Negation)
¬
(
¬
A
) =
A
.
(Complementary Property)
A
∩
¬
A
=
∅
,
A
∪
¬
A
=
U
, where
U
denotes the universe.
Theorem: (a)
A
∩
B
⊂
A
. (b)
A
⊂
A
∪
B
. (c)
A
∩
∅
=
∅
. (d)
A
∪
∅
=
A
. (e)
A
⊂
B
iff
¬
B
⊂
¬
A
(f)
A
⊂
B
iff
A
∩
¬
B
=
∅
. (g)
A

B
=
A
∩
¬
B
. (h) (
A
⊂
B
and
B
⊂
C
) implies
A
⊂
C
. (i)
A
⊂
B
implies
A
∪
C
⊂
B
∪
C
. (j)
A
⊂
B
implies
A
∩
C
⊂
B
∩
C
.
(a)
If
A
∩
B
=
C
, then
C
⊂
A
and
C
⊂
B
.
Since
C
=
A
∩
B
 (1) by assumption, and
A
∩
B
⊂
A
 (2)
by the above
theorem (a), so substituting (1) into (2) proves
C
⊂
A
.
Similarly, since
A
∩
B
=
B
∩
A
⊂
B
 (3) by the commutative law and by the above theorem (a),
substituting (1) into (3) yields
C
⊂
B
.
(b)
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 Spring '09
 Addition, Commutativity, Associativity, Associative Law

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