Question

Suppose \(A=\\{0,2,4,6,8\\}, B=\\{1,3,5,7\\}\) and \(C=\\{2,8,4\\} .\) Find: (a) \(A \cup B\) (b) \(A \cap B\) (c) \(A-B\) (d) \(A-C\) (e) \(B-A\) (f) \(A \cap C\) (g) \(B \cap C\) (h) \(C-A\) (i) \(C-B\)

Short answer

\(A \cup B = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\), \(A \cap B = \emptyset\), \(A - B = \{0, 2, 4, 6, 8\}\), \(A - C = \{0, 6\}\), \(B - A = \{1, 3, 5, 7\}\), \(A \cap C = \{2, 4, 8\}\), \(B \cap C = \emptyset\), \(C - A = \emptyset\), \(C - B = \{2, 8, 4\}\)

Step-by-step solution

Set A Union B

To find \(A \cup B\), combine all the elements from both sets A and B. So, \(A \cup B = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\).

Intersection of A and B

The intersection of two sets is the set of elements that are common to both sets. There are no common elements in set A and B, thus \(A \cap B = \emptyset\).

Relative Complement of B in A

Set \(A - B\) means we should include in the resultant set the elements that are in A, but not in B. So, \(A - B = A = \{0, 2, 4, 6, 8\}\).

Relative Complement of C in A

Set \(A - C\) means we should include in the resultant set the elements that are in A, but not in C. So \(A - C = \{0, 6\}\).

Relative Complement of A in B

Set \(B - A\) means we should include in the resultant set the elements that are in B, but not in A. So \(B - A = B = \{1, 3, 5, 7\}\).

Intersection of A and C

The intersection of two sets is the set of elements that are common to both sets. So, \(A \cap C = \{2, 4, 8\}\).

Intersection of B and C

The intersection of two sets is the set of elements that are common to both sets. Here, there are no common elements in sets B and C, thus \(B \cap C = \emptyset\).

Relative Complement of A in C

Set \(C - A\) means we should include in the resultant set the elements that are in C, but not in A. So \(C - A = \emptyset\).

Relative Complement of B in C

Set \(C - B\) means we should include in the resultant set the elements that are in C, but not in B. So \(C - B = C = \{2, 8, 4\}\).