Ask a New Question
Menu

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

1 answer

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)