Question

Let U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 7}, and B = {3, 4, 5}. Find the set (A ∪ B)′ .Select the correct choice below and, if necessary, fill in the answer box tocomplete your choice.A. (A ∪ B)′ =(Use a comma to separate answers as needed. Use ascending order.)B. (A ∪ B)′ is the empty set.

Answer 1

Given the set U

`U=\mleft\lbrace1,2,3,4,5,6,7\mright\rbrace`

And the subsets:

`\begin{gathered} A=\mleft\lbrace1,3,5,7\mright\rbrace \\ B=\mleft\lbrace3,4,5\mright\rbrace \end{gathered}`

The set (A∪B) represents the union between both sets A and B and is represented by the elements of these sets combined.

Then, the set (A∪B)', represents the complement of the set (A∪B), and will include elements of U that are not included in the set (A∪B)

The elements that represent the union of A and B are:

`\begin{gathered} A=\mleft\lbrace1,3,5,7\mright\rbrace \\ B=\mleft\lbrace3,4,5\mright\rbrace \\ (A\cup B)=\mleft\lbrace1,3,4,5,7\mright\rbrace \end{gathered}`

Then the complement of (A∪B) comprehends all elements of U not included in this set:

U= {1, 2, 3, 4, 5, 6, 7}

(A∪B)= {1, 3, 4, 5, 7}

The elements of U not included in (A∪B) are "2" and "6", so these events would be included in its complement (A∪B)'

`(A\cup B)^(\prime)=\mleft\lbrace2,6\mright\rbrace`