Using properties of sets show that (i) \(A ∪ (A ∩ B) = A \) (ii) \(A ∩ (A ∪ B) = A.\)
Solution and Explanation
(i) To show: \(A ∪ (A ∩ B) = A \)
We know that
\(A ⊂ A \)
\(A ∩ B ⊂ A \)
\(∴ A ∪ (A ∩ B) ⊂ A … (1) \)
Also, \(A ⊂ A ∪ (A ∩ B) … (2) \)
∴ From (1) and (2) ,\( A ∪ (A ∩ B) = A \)
(ii) To show: \(A ∩ (A ∪ B) = A\)
\(A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B) \)
\(= A ∪ (A ∩ B)\)
\(= A\) {from (1)}
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Concepts Used:
Operations on Sets
Some important operations on sets include union, intersection, difference, and the complement of a set, a brief explanation of operations on sets is as follows:
1. Union of Sets:
- The union of sets lists the elements in set A and set B or the elements in both set A and set B.
- For example, {3,4} ∪ {1, 4} = {1, 3, 4}
- It is denoted as “A U B”
2. Intersection of Sets:
- Intersection of sets lists the common elements in set A and B.
- For example, {3,4} ∪ {1, 4} = {4}
- It is denoted as “A ∩ B”
3.Set Difference:
- Set difference is the list of elements in set A which is not present in set B
- For example, {3,4} - {1, 4} = {3}
- It is denoted as “A - B”
4.Set Complement:
- The set complement is the list of all elements present in the Universal set except the elements present in set A
- It is denoted as “U-A”