Define the sets A, B, C, and D as follows:

A = {-3, 0, 1, 4, 17}

B = {-12, - 5, 1, 4, 6}

C = {x ∈ Z: x is odd}

D = {x ∈ Z: x is positive}

For each of the following set expressions, if the corresponding set is finite, express the set using roster notation. Otherwise, indicate that the set is infinite.

(a)

A ∪ B

(b)

A ∩ B

(c)

A ∩ C

(d)

A ∪ (B ∩ C)

(e)

A ∩ B ∩ C

(f)

A ∪ C

(g)

(A ∪ B) ∩ C

(h)

A ∪ (C ∩ D)

(a) A∪B = {-12,-5,-3, 0, 1, 4, 6, 17}

(b) A∩B = {1,4}

(c) A∩C = {-3,1,17}

(d) A ∪ (B ∩ C) = {-5,-3,0,1,4,17}

(e) A ∩ B ∩ C = {1}

(f) A ∪ C, the set is infinite.

(g) (A ∪ B) ∩ C = {-5,-3,1,17}

(h) A ∪ (C ∩ D), the set is infinite.

Step-by-step explanation:

(a) Recall that the union of A and B is the set that contains all the elements that are in A or in B, and that in set notation no repetitions are allowed, so we can only write 1 once.

(b) Recall the the intersection of A and B is the set formed by the elements that are in A and B at the same time. In this case, only 1 and 4 satisfy that condition.

(c) In this case we list only the elements in A that are odd.

(d) In this case we perform first the operation B ∩ C={-5}. Then, we perform A∪{-5}.

(e) Recall that set operations are associative, so A ∩ B ∩ C = (A ∩ B) ∩ C. As we have calculated A∩B = {1,4}, we only need to find the odd numbers, which is only 1.

(f) The set is infinite because C is infinite. Recall that the union of an infinite set with any other, is infinite too. In plain words, when we perform a union of set we are adding elements.

(g) These are the elements that are in A ∪ B and are odd too.

(h) Notice that the set C ∩ D is infinite, because is formed by the positive odd integers. So, its union with any other set is infinite too.