Assignment:
Q1. Let T = {m ∈Z|m = 1 + (-1)i, for some integer i}. Describe T.
Q2. Let A = {m ∈Z|m = 2i - 1, for some integer i},
B = {n ∈Z|n = 3j + 2, for some integer j},
C = {p ∈Z|p = 2r + 1, for some integer r}, and
D = {q ∈Z|q = 3s – 1, for some integer s}.
a) Describe the 4 sets (by enumerating their elements).
b) Is A = D? Explain.
c) Is B = D? Explain.
Q3. Let R, S, and T be defined as follows:
R = {x ∈Z|x is divisible by 2},
S = {y ∈Z|y is divisible by 3},
T = {z ∈Z|z is divisible by 6}.
a) Describe the 3 sets (by enumerating their elements).
b) Is T ⊆ S? Explain.
c) Find R ∩ S. Explain.
Q4. Let A = {a, b, c}, B = {b, c, d}, and C = {b, c, e}.
a) Find A ∩( B U C), (A ∩ B) U C, and (A ∩ B) U (A ∩ C). Which of these sets are equal?
b) Find (A - B) - C and A - (B - C). Are these sets equal?
Q5. For all sets A, B, and C,
(A - B) ∩ (C - B) = A - (B U C).
Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
Q6. For all sets A, B, and C,
if A ∩ C ⊆ B ∩ C and A U C ⊆ B U C, then A = B.
Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
Q7. For all sets A, B, and C,
(A U B) ∩ C = A U (B ∩ C).
Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
Q8. Derive the following property:
For all sets A, B, and C,
(A - B) - C = (A - C) - B.
Q9. Derive the following property:
For all sets A and B,
A - (A - B) = A ∩ B.
Q10. Suppose A, B, and C are sets.
a) Are A - B and B - C necessarily disjoint? Explain.
b) Are A - B and C - B necessarily disjoint? Explain
c) Are A - (B U C) and B - (A U C) necessarily disjoint? Explain.
Q11. Let S = {a, b, c} and for each integer i = 0, 1, 2, 3, let Si be the set of all subsets of S that have i elements. List the elements in S0, S1, S2, and S3. Is {S0, S1, S2, S3} a partition of ℘(S) (the power set of S)?
Provide complete and step by step solution for the question and show calculations and use formulas.