Question

Let A,B,C are subsets of set X. Then consider the validity of the following set theoretic statement:

• A. \(A\cup(B/C)=(A\cup B)/(A\cup C)\)
• B. \((A/B)/C=A/(B\cup C)\)
• C. \((A\cup B)=A/B\)
• D. \(A/B=B/C\)

Answer 1

The Correct Option is B

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): $$(A/B)/C = A/(B \cup C)$$