CSCI 2400/SENG 1010
Solutions for Practice Questions Set 1028

1. Write an enumeration of the members of set {x | x is an integer and x ≥ 0 and x > x² − 5}.

   {0, 1, 2}

2. Give an enumeration of the members of each of the following sets.

   1. {1, 3, 5, 6} ∪ {2, 3, 5}

      {1, 2, 3, 5, 6}

   2. {1, 3, 5, 6} ∩ {2, 3, 5, 9}

      {3, 5}

   3. {1, 3, 5, 6} − {2, 3, 5, 9}

      {1, 6}

3. What is |{2, 2, 3, 3}|? Read the question carefully.

   2

4. Is it always true that |A ∪ B| = |A| + |B|? If so, give an argument for why it is true. If not, give a counterexample.

   No, it is not always true. Suppose A = {1, 2, 3} and B = {1, 2, 3}. Then |A| = 3 and |B| = 3. But A ∪ B = {1, 2, 3}, which has cardinality 3.

5. True or false?

   1. {2, 4, 6} ⊆ {2, 4, 6, 8}

      True

   2. {2, 4, 6} ∈ {2, 4, 6, 8}

      False

   3. S − S = { } for every set S

      True

   4. 2 ∈ {2}

      True

   5. { } ∈ { }

      False

   6. { } ⊆ { }

      True

6. What are sets A and B if A − B = {1, 5, 7, 8}, B − A = {2, 10} and A ∩ B = {3, 6, 9}?

   A = {1, 3, 5, 6, 7, 8, 9}
   B = {2, 3, 6, 9, 10}

7. Suppose that A, B and C are sets. Can you conclude that A = B if

   1. A ∪ C = B ∪ C?

      No. Suppose A = {1}, B = {2} and C = {1, 2}. Then A ∪ C = {1, 2} and B ∪ C = {1, 2}, but A ≠ B.

   2. A ∩ C = B ∩ C?

      No. Suppose A = {1}, B = {2} and C = {3}. Then A ∩ C = { } and B ∩ C = { }, but A ≠ B.

   3. A ∪ C = B ∪ C and A ∩ C = B ∩ C?

      Yes.

      For any set S, S = (S ∩ C) ∪ (S − C). That is, S consists of the members of S that are shared with C together with the members of S that are not shared with C.

      Also S − C = (S ∪ C) − C, since all members of C are being removed from S ∪ C. So S = (S ∩ C) ∪ ((S ∪ C) − C).

      Suppose A ∩ C = B ∩ C = R and A ∪ C = B ∪ C = T. Then
      

      A	=	((A ∩ C) ∪ ((A ∪ C) − C)
      	=	(R ∪ (T − C)
      B	=	((B ∩ C) ∪ ((B ∪ C) − C)
      	=	(R ∪ (T − C)

      So A = B.