Step by step
The question
Question: find (A ∪ B)′, when ξ is the numbers 1 to 10, A is {1, 2, 5, 10} and B is {1, 3, 5, 7, 9}.
ξ = {1, 2, …, 10}
A = {1, 2, 5, 10}
B = {1, 3, 5, 7, 9}
1 Read the sets
A set is a list of numbers in curly brackets.
Each number is an element.
ξ is the set of everything we look at.
ξ = everything we look at
∪ = in A or in B (or both)
A′ = not in A
(A ∪ B)′ means: the elements in neither A nor B.
(A ∪ B)′ = = in neither A nor B
2 Draw the Venn diagram
3 Shade the part
(A ∪ B)′ is in neither A nor B.
On the diagram: outside the circles.
4 Read the answer
Careful!
The complement is not the other set.
It is everything in ξ that is not in the set.
Careful: ′ = everything in ξ that is NOT in the set
The shaded part holds 4, 6, 8.
We list them in order, each once.
(A ∪ B)′ = {4, 6, 8}
5 Check
Let's check by counting.
Everything, 10, minus the 7 inside: 3.
The same count.
So it is right!
n(ξ) − n(A ∪ B) = 10 − 7 = 3
Let's look back.
First: read the sets.
Then: draw the Venn diagram.
Then: shade the part.
Then: read the answer.
Last: check.
Answer
(A ∪ B)′ = {4, 6, 8}