Question

Suppose \(A=\\{4,3,6,7,1,9\\}, B=\\{5,6,8,4\\}\) and \(C=\\{5,8,4\\} .\) 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) \(B \cup C\) (i) \(C-B\)

Short answer

(a) \(A\cup B = \{1, 3, 4, 5, 6, 7, 8, 9\} \n\) (b) \(A\cap B = \{4,6\} \)\n(c) \(A - B = \{1,3,7,9\} \) (d)\( A - C = \{1,3,6,7,9\}\) (e) \(B - A = \{5,8\}\) (f)\( A\cap C = \{4\}\) (g)\(B\cap C = \{4,5,8\} \) (h) \(B\cup C = \{4,5,6,8\}\) (i) \(C - B= \{\}\)

Step-by-step solution

Definitions

Set operations include union, intersection, and subtraction. The union of two sets A and B, denoted by \(A\cup B\), is the set of elements which are in A, in B, or in both A and B. The intersection of two sets A and B, denoted by \(A\cap B\), is the set of all objects that are members of both the sets .. The subtraction of two sets A and B, denoted by \(A - B\), is the set of elements which are only in A but not in B. Now, these operations can be applied to the given sets A, B and C.

Union of A and B

The union of sets A and B, \(A\cup B\), includes all the numbers that are in A, in B, or in both. That gives us: \(\{1,3,4,5,6,7,8,9\}\).

Step 3:Intersection of A and B

The intersection of sets A and B,\(A\cap B\), contains all numbers that are common to both A and B. That gives us: \(\{4,6\}\).

Step 4:Subtraction A - B

To find the subtraction A - B, we take all the elements in A that are not in B. That gives us: \(\{1,3,7,9\}\).

Step 5:Subtraction A - C

To find the subtraction A - C, we take all the elements in A that are not in C. That gives us: \(\{1,3,6,7,9\}\).

Step 6:Subtraction B - A

To find the subtraction B - A, we take all the elements in B that are not in A. That gives us: \(\{5,8\}\).

Step 7:Intersection A and C

The intersection of sets A and C,\(A\cap C\), contains all numbers that are common to both A and C. That gives us: \(\{4\}\).

Step 8:Intersection B and C

The intersection of sets B and C,\(B\cap C\), contains all numbers that are common to both B and C. That gives us: \(\{4,5,8\}\).

Step 9:Union B and C

The union of sets B and C, \(B\cup C\), includes all the numbers that are in B, in C, or in both. That gives us: \(\{4,5,6,8\}\).

Step 10:Subtraction C - B

To find the subtraction C - B, we take all the elements in C that are not in B. In this case, C and B have the same elements, so the resulting set is empty: \(\{\}\).