Step by step
The question
Question: find n(A ∪ B), when ξ is the numbers 1 to 15, A is {3, 6, 9, 12, 15} and B is {5, 10, 15}.
ξ = {1, 2, …, 15}
A = {3, 6, 9, 12, 15}
B = {5, 10, 15}
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)
n( ) = how many elements
n(A ∪ B) means: the elements in A or in B, or in both.
n means: how many elements.
n(A ∪ B) = = in A or in B, or in both
2 Draw the Venn diagram
3 Shade the part
n(A ∪ B) is in A or in B, or in both.
On the diagram: A only, the middle, B only.
4 Read the answer
We count the shaded elements: 3, 5, 6, 9, 10, 12, 15.
So n(A ∪ B) = 7.
n(A ∪ B) = 7
5 Check
Let's check by counting.
A has 5, B has 3, and 1 were counted twice.
5 plus 3 minus 1 is 7: the same count.
So it is right!
n(A) + n(B) − n(A ∩ B) = 5 + 3 − 1 = 7
Let's look back.
First: read the sets.
Then: draw the Venn diagram.
Then: shade the part.
Then: read the answer.
Last: check.
Answer
n(A ∪ B) = 7