Step by step
The question
Question: find A ∪ B, when ξ is the numbers 1 to 10, A is {2, 4, 6, 8} and B is {1, 2, 3, 4}.
ξ = {1, 2, …, 10}
A = {2, 4, 6, 8}
B = {1, 2, 3, 4}
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 ∪ B means: the elements in A or in B, or in both.
A ∪ B = = in A or in B, or in both
2 Draw the Venn diagram
3 Shade the part
A ∪ B is in A or in B, or in both.
On the diagram: A only, the middle, B only.
4 Read the answer
Careful!
2 and 4 are in both sets.
In a set, each element is written once.
Careful: not {2, 4, 6, 8, 1, 2, 3, 4} ✗
The shaded part holds 1, 2, 3, 4, 6, 8.
We list them in order, each once.
A ∪ B = {1, 2, 3, 4, 6, 8}
5 Check
Let's check by counting.
A has 4, B has 4, and 2 were counted twice.
4 plus 4 minus 2 is 6: the same count.
So it is right!
n(A) + n(B) − n(A ∩ B) = 4 + 4 − 2 = 6
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 = {1, 2, 3, 4, 6, 8}