Step by step
The question
Question: find (A ∩ B)′, when ξ is the numbers 1 to 12, A is {1, 2, 3, 6} and B is {1, 2, 4, 8}.
ξ = {1, 2, …, 12}
A = {1, 2, 3, 6}
B = {1, 2, 4, 8}
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 both
A′ = not in A
(A ∩ B)′ means: the elements not in both.
(A ∩ B)′ = = not in both
2 Draw the Venn diagram
3 Shade the part
(A ∩ B)′ is not in both.
On the diagram: A only, B only, 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 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
We list them in order, each once.
(A ∩ B)′ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
5 Check
Let's check some elements against the rule: not in both.
They fit.
So it is right!
3: in A, not in B; 4: not in A, in B
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)′ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}