Question

Given two sets A and B, the intersection of these sets, denoted A ∩ B, is...

  • Given two sets A and B, the intersection of these sets, denoted A ∩ B, is the set containing the elements that are in both A and B. That is, A ∩ B = {x : x ∈ A and x ∈ B}.
  • Two sets A and B are disjoint if they have no elements in common. That is, if A ∩ B = ∅.
  • Given two sets A and B, the union of these sets, denoted A ∪ B, is the set containing the elements that are either in A or in B (or in both). That is, A ∪ B = {x : x ∈ A or x ∈ B}.
  • Given two sets A and B, the relative complement or difference of these sets, denoted B −A, is the set of elements that are in B but not in A. That is, B − A = {x : x ∈ B and x /∈ A}.

  • Given a universal set U and a set A, the complement of the set A with respect to the set U, denoted as AC or A, is

the set U − A. That is, A = {x ∈ U : x /∈ A}.

  • Given two sets A and B, the symmetric difference of these sets, denoted A ⊕ B, is the set of elements that are in A or B but not both. That is, A ⊕ B = (A ∪ B) − (A ∩ B).
  1. Let A = {a,b,c,d,e,f}, B = {a,c,e,f,g,h,i}, C = {b,c,d,f,g,h} and U = {a,b,c,d,e,f,g,h,i,j}. Express each of the following in roster notation.

              (a) A ∪ B                                                      (b) A ∩ B                                                      (c) A ⊕ B

              (d) A − C                                                       (e) C − A                                                       (f) C

               (g) A − (B ∪ C)                                            (h) A ∪ (B ∩ C)                                (i) (A ∪ B) ∩ (A ∪ C)

  1. Suppose U = {1,2,3,4,5,6} and B = {2,3,4}.
    1. What can you say about the set A if A ∪ B = {1,2,3,4,5}?
    2. What can you say about the set A if A ∩ B = {2,3}?
    3. What can you say about the set A if A ⊕ B = {1,2,4,5}?
  2. What can you day about the sets A and B if A ⊕ B = ∅?
  3. Draw a general Venn diagram representing each of the following:

  1. A ∩ B (b) A − B

               (c) (A ∪ B) − C                                                                                (d) A ∪ (B − C)

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