# Set Theory questions

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)?

#### Solution Preview

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.