The sets A - B, B - A and A ∩ B are mutually disjoint sets. Use example to observe if this is true.

To determine whether the sets A - B, B - A, and A ∩ B are mutually disjoint, let's first clarify what these sets represent. The notation A - B refers to the elements that are in set A but not in set B, while B - A includes the elements in set B but not in set A. The intersection A ∩ B consists of elements that are present in both sets A and B. For these sets to be mutually disjoint, they must not share any elements with one another.

Defining the Sets

Let's define two sets for our example:

• A = {1, 2, 3, 4}
• B = {3, 4, 5, 6}

Calculating the Sets

Now, we can calculate A - B, B - A, and A ∩ B:

• A - B = {1, 2} (elements in A that are not in B)
• B - A = {5, 6} (elements in B that are not in A)
• A ∩ B = {3, 4} (elements common to both A and B)

Analyzing Mutual Disjointness

Next, we need to check if these sets are mutually disjoint. This means that no set should share any elements with the others:

• Elements in A - B: {1, 2}
• Elements in B - A: {5, 6}
• Elements in A ∩ B: {3, 4}

Now, let's see if there are any common elements:

• A - B (1, 2) has no elements in common with B - A (5, 6).
• A - B (1, 2) has no elements in common with A ∩ B (3, 4).
• B - A (5, 6) has no elements in common with A ∩ B (3, 4).

Conclusion on Mutual Disjointness

Since none of the sets A - B, B - A, and A ∩ B share any elements, we can conclude that they are indeed mutually disjoint. This example illustrates that when you have two sets, the differences and the intersection can be distinctly separated without overlap.

Generalization

This property holds true for any two sets A and B. Whenever you compute A - B, B - A, and A ∩ B, you will find that these sets do not share elements, confirming their mutual disjointness. This concept is fundamental in set theory and helps in understanding how different sets interact with one another.