Understanding the Equivalence of A∪B A∩B: A B

In set theory, the relationship between the union (A∪B) and intersection (A∩B) of two sets A and B can be a powerful tool for determining when these sets are equal. Specifically, we will prove that if A∪B A∩B, then A B, and vice versa. This is a fundamental concept in set theory and can be useful in various mathematical contexts.

Proving the Forward Direction: A∪B A∩B implies A B

To prove the forward direction, we start by assuming that A∪B A∩B, and then we need to show that this implies A B. This can be broken down into two parts: proving that A is a subset of B (A?B) and that B is a subset of A (B?A).

Proving A?B:

Let x be an element of A (x∈A). Since x is in A, x is also in A∪B (A is a subset of A∪B). Given A∪B A∩B, it follows that x must also be in A∩B. If x is in A∩B, then x is in both A and B.

This shows that x∈B, and thus A?B.

Proving B?A:

Now, let y be an element of B (y∈B). Since y is in B, y is also in A∪B (B is a subset of A∪B). Given A∪B A∩B, it follows that y must also be in A∩B. If y is in A∩B, then y is in both A and B.

This shows that y∈A, and thus B?A.

Conclusion of the Forward Direction:

Since A?B and B?A, it follows that A B.

Proving the Reverse Direction: A B implies A∪B A∩B

To prove the reverse direction, we assume that A B, and then we need to show that this implies A∪B A∩B.

Given that A B, we can substitute B with A in the union and intersection expressions:

Substitution:

A∪B A∪A A

A∩B A∩A A

This shows that A∪B A∩B.

Conclusion of the Reverse Direction:

We have proven that if A B, then A∪B A∩B.

Common Misunderstandings and Clarifications

The implication "if A∪B A∩B then A B" can be misleading if we don't consider the empty set. To clarify, let us examine some special cases:

Case 1: B is the Null Set (?) and A is a Non-Null Set:

If B ? and A is a non-null set, then:

A∩B ? (the intersection of any set with the empty set is empty). A∪B A (the union of any set with the empty set is the set itself). Thus, A∪B ≠ A∩B.

This confirms that the statement only holds when both A and B are non-null sets.

Case 2: A is the Null Set (?) and B is a Non-Null Set:

If A ? and B is a non-null set, then:

A∩B ? (the intersection of the empty set with any set is empty). A∪B B (the union of the empty set with any set is that set). Thus, A∪B ≠ A∩B.

Again, this shows that the statement only holds when both A and B are non-null sets.

General Case: Both A and B are Non-Null Sets:

If both A and B are non-null sets, and A∪B A∩B:

A∩B is a subset of both A and B. A∪B is the union of A and B.

Given A∪B A∩B, it must be that the elements in A but not in B, and the elements in B but not in A are both empty sets. This implies that A and B must be equal sets, as there are no elements in either set that are not present in the other.

Conclusion

We have proven both directions, establishing that A∪B A∩B if and only if A B, except in the case where either A or B is the null set. The key to understanding this concept lies in recognizing the properties of union and intersection and how they behave with respect to set equality.