Let A,B,C are subsets of set X. Then consider the validity of the following set theoretic statement:
- \(A\cup(B/C)=(A\cup B)/(A\cup C)\)
\((A/B)/C=A/(B\cup C)\)
- \((A\cup B)=A/B\)
- \(A/B=B/C\)
The Correct Option is B
Solution and Explanation
We are given the identity: $$(A / B) / C = A / (B \cup C)$$
Let's understand what this means using set theory:
- $A / B$ means the set of elements in A but not in B: $$A \setminus B$$
- So, $(A / B) / C = (A \setminus B) \setminus C$
Now, we recall the property of set difference:
$$(A \setminus B) \setminus C = A \setminus (B \cup C)$$
This is a well-known identity in set theory.
Therefore, we conclude:
$$(A / B) / C = A / (B \cup C)$$
So the correct answer is: Option (B): $$(A/B)/C = A/(B \cup C)$$
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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”