Question

Let the universal set U = {a, b, c, d} with sets A = {a, b} and B = {b, c}. Find the set A ∪ BC.Which of the sets below are disjoint to this set? (Select all that apply.)[Hint: For each of the sets below, find the elements of the set and compare to the set A ∪ BC in order to determine if they are disjoint.]a) A ∩ Bb) A ∩ B^Cc) A^C ∩ Bd) A ∪ Be) A ∪ B^Cf) A^C ∪ Bg) A^C ∩ B^Ch) A^C ∪ B^Ci) None are disjoint

Answer 1

`A^c\cap B`

Explanation:

Given,

U = {a, b, c, d}, A = {a, b} and B = {b, c},

`A^c=U-A=\{c, d\}`

`B^c=U-B = \{ a, d\}`

Thus,

`A\cup B^c=\{a, b, d\}`

`A\cap B = \{b\}`

`A\cap B^c = \{a\}`

`A^c\cap B = \{c\}`

`A\cup B = \{a, b, c, d\}`

`A^c\cup B = \{b, c, d\}`

`A^c\cap B^c = \{d\}`

Now, two sets are called disjoint if there intersection is ∅,

By the above explanation it is clear that,

`(A\cup B^c)\cap (A^c\cap B) = \phi`

Hence,
`(A^c\cap B)` is disjoint to
`(A\cap B^c)`