Description
Problem 1. (25 points)
Suppose that A, B, and C are sets. Prove or disprove that (A − B) − C = (A − C) − B.
Solution.
Suppose A = {1, 2, 3}
B = {1, 2, 3, 4}
C = {1}
Then B − C = {2, 3, 4}
A − (B − C) = 1
Also A − B = ∅
Then (A − B) − C = ∅
∴ (A − B) − C 6= (A − C) − B
1
Problem 2. (25 points)
Determine whether the symmetric difference is associative; that is, if A, B, and C are sets, does it follow
that A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C?
a. Use a Venn diagram.
b. Use a membership table.
c. Use set identities.
Solution.
a. Use a Venn diagram.
b. Membership table
A B C A ⊕ B B ⊕ C (A ⊕ B) ⊕ C A ⊕ (B ⊕ C)
0 0 0 0 0 0 0
0 0 1 0 1 1 1
0 1 0 1 1 1 1
0 1 1 1 0 0 0
1 0 0 1 0 1 1
1 0 1 1 1 0 0
1 1 0 0 1 0 0
1 1 1 0 0 1 1
∴ A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C because they have the same truth values
c. Use set identities.
2
Problem 3. (25 points)
Determine whether f is a function from Z to R if
a. f(n) = ±n
b. f(n) = ln
2
m
c. f(n) = √
n2 + 1
d. f(n) = √
n
e. f(n) = 1
n2 − 4
Solution.
a. f(n) = ±n
b. f(n) = ln
2
m
c. f(n) = √
n2 + 1
d. f(n) = √
n
e. f(n) = 1
n2 − 4
3
Problem 4. (25 points)
Consider the function f : Z → (N − {0}) where f(n) =
1 − 2n n ≤ 0
2n n > 0
a. Prove that f is a bijection by showing that it is both injective and surjective.
b. Find the inverse function f
−1
.
Solution.
a. Prove that f is a bijection by showing that it is both injective and surjective.
b. Find the inverse function f
−1
.
4
Aggie Honor Statement: On my honor as an Aggie, I have neither given nor received any unauthorized
aid on any portion of the academic work included in this assignment.
Checklist: Did you…
1. abide by the Aggie Honor Code?
2. solve all problems?
3. start a new page for each problem?
4. show your work clearly?
5. type your solution?
6. submit a PDF to eCampus?
5
Powered by TCPDF (www.tcpdf.org)
