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