Question

let the universal set = (a,b,c,d,e), X =(a,c,e), Y= (a,b,c). find:
1. X-Y
2. Y-X
3. hence verify that (X-Y)intersect (Y-X) is empty set

Answer 1

To solve the problem, we first need to identify the universal set \( U \), and the sets \( X \) and \( Y \).

- Universal set \( U = \{a, b, c, d, e\} \)
- Set \( X = \{a, c, e\} \)
- Set \( Y = \{a, b, c\} \)

### 1. \( X - Y \)

The set difference \( X - Y \) consists of elements that are in \( X \) but not in \( Y \).

- Elements in \( X \): \( a, c, e \)
- Elements in \( Y \): \( a, b, c \)

So, removing any elements from \( X \) that are also in \( Y \):

- \( X - Y = \{a, c, e\} - \{a, b, c\} = \{e\} \)

### 2. \( Y - X \)

The set difference \( Y - X \) consists of elements that are in \( Y \) but not in \( X \).

- Elements in \( Y \): \( a, b, c \)
- Elements in \( X \): \( a, c, e \)

So, removing any elements from \( Y \) that are also in \( X \):

- \( Y - X = \{a, b, c\} - \{a, c, e\} = \{b\} \)

### 3. Verify that \( (X - Y) \cap (Y - X) \) is the empty set

We have:

- \( X - Y = \{e\} \)
- \( Y - X = \{b\} \)

Now we find the intersection \( (X - Y) \cap (Y - X) \):

- \( (X - Y) \cap (Y - X) = \{e\} \cap \{b\} = \{\} \)

Since there are no common elements in \( \{e\} \) and \( \{b\} \), the intersection is indeed the empty set.

### Conclusion

1. \( X - Y = \{e\} \)
2. \( Y - X = \{b\} \)
3. \( (X - Y) \cap (Y - X) = \emptyset \) (the empty set)