Let A and B be two non-empty subsets of set X such that A is not a subset of B, then which of the following is always true?a)A is a subset of the complement of Bb)B is a subset of Ac)A and B are disjointd)A and the complement of B are non-disjointCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation:

To understand why option D is always true, let's analyze the given information step by step.

1. A is not a subset of B:

This means that there exists at least one element in A that is not in B. Let's call this element 'x'.

2. A is a subset of the complement of B:

To prove this, we need to show that every element in A is also in the complement of B.

- Let's assume that there exists an element 'y' in A that is not in the complement of B. This would mean that y is in B.
- Since A is not a subset of B (as stated in the question), there must be an element 'x' in A that is not in B.
- But we already know that x is not in B, so y cannot be equal to x.
- Therefore, we have found an element 'y' in A that is not in the complement of B, which contradicts our assumption.
- Hence, every element in A must be in the complement of B, making A a subset of the complement of B.

3. A and the complement of B are non-disjoint:

To prove this, we need to show that there exists at least one element that is common to both A and the complement of B.

- Since A is not a subset of B, there exists an element 'x' in A that is not in B.
- This means that x must be in the complement of B.
- Therefore, x is a common element to both A and the complement of B.
- Hence, A and the complement of B are non-disjoint.

Conclusion:

Based on the above analysis, we can conclude that option D is always true. A is a subset of the complement of B, and A and the complement of B are non-disjoint.

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