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