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Set Theory questions

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1. Let T = {m   m = 1 + (-1)i, for some integer i}. Describe T.
2. Let A = {m   m = 2i - 1, for some integer i},
B = {n   n = 3j + 2, for some integer j},
C = {p   p = 2r + 1, for some integer r}, and
D = {q   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.
3. Let R, S, and T be defined as follows:
R = {x    x is divisible by 2},
S = {y    y is divisible by 3},
T = {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.
4. Let A = {a, b, c}, B = {b, c, d}, and C = {b, c, e}.
a) Find A  ( B  C), (A  B)  C, and (A  B)  (A  C). Which of these sets are equal?
b) Find (A  B)  C and A  (B  C). Are these sets equal?
5. For all sets A, B, and C,
(A  B)  (C  B) = A  (B  C).
Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
6. For all sets A, B, and C,
if A  C  B  C and A  C  B  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.
7. For all sets A, B, and C,
(A  B)  C = A  (B  C).
Either prove it is true (from the definitions of the set operations) or find a counterexample to show that is false.
8. Derive the following property:
For all sets A, B, and C,
(A  B)  C = (A  C)  B.
9. Derive the following property:
For all sets A and B,
A  (A  B) = A  B.
10. 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  C) and B  (A  C) necessarily disjoint? Explain.
11. 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)?

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1. Let T = {m   m = 1 + (-1)i, for some integer i}. Describe T.

Solution:

When i = 0
m = 1 + (-1)0 = 1+1 = 2

When i is odd integer
m = 1+(-1) = 0

When i is even integer
m = 1+1 = 2

Thus, T = {0, 2}

2. Let A = {m   m = 2i - 1, for some integer i},
B = {n   n = 3j + 2, for some integer j},
C = {p   p = 2r + 1, for some integer r}, and
D = {q   q = 3s - 1, for some integer s}.
a) Describe the 4 sets (by enumerating their elements).

Solution:

A = {--------------------- -7, -5, -3, -1, 1, 3, 5, 7,...................}
B = {----------------------7, -4, -1, 2, 5, 8, 11,.......................}
C = {--------------------- -7, -5, -3, -1, 1, 3, 5, 7,...................}
D = {----------------------7, -4, -1, 2, 5, 8, 11,.......................}

b) Is A = D? Explain.

Solution:

From part (a), we can note that A≠D as the elements in the sets A and D are not the same.

Answer: NO

c) Is B = D? Explain.

Solution:
From part (a), we can note that B=D as the elements in the sets B and D are the same.

Another proof:
Let n B
Then n = 3j + 2
We can write it as
n = 3j+3-1 = 3(j+1)-1
let s = j+1
Thus, n = 3s-1, where s is some integer

This means n D
This gives ---------------------(i)

Now, let m D
Then m = 3s-1
We can write it as
m = 3s-3+2 = 3(s-1)+2
Let j = s-1
Thus, m = 3j+2

This means m B
This gives ---------------------(ii)

Now, from (i) and (ii)
B = D

Answer: YES

3. Let R, S, and T be defined as follows:
R = {x    x is divisible by 2},
S = {y    y is divisible by 3},
T = {z    z is ...

Solution Summary

This posting explains how to prove or disprove statements using counterexamples in set theory.

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