The complement of a set

Key points

An image of a Venn diagram. The Venn diagram has been used to classify shapes. One circle represents, triangles, the other circle represents, blue shapes. — Image caption,
The complement of set B, B’, contains all the elements that are not in set B.

The of a contains all the that are not in the set. A set plus the complement of a set is the .
On a the complement of a set is the outside the circle that represents the set.
\(P’\) is the complement of set \(P\).
The intersection of one set and the complement of another set identifies elements with the attributes of one set and none of the attributes of the other set. This can be used to classify shapes and to support probability calculations.

Understanding the complement of a set

To list the elements of the complement of set \(P\):
- \(P'\) contains all the elements that are not in set \(P\). These are the elements in the universal set that are not in set \(P\).
To identify the complement of a set \(P\) on a Venn diagram:
- \(P'\) is the region outside the circle that represents set \(P\).

Examples

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Image caption,
Describe 𝑩’, the complement of set 𝑩, and list the elements in 𝑩’.
Image caption,
𝑩 = {blue shapes}. The complement of set 𝑩 is 𝑩’. 𝑩’ is the set of shapes that are not blue.
Image caption,
The elements of 𝑩’ can be listed. These are all the shapes that are in the universal set (ξ) that are not in set 𝑩.
Image caption,
The Venn diagram shows shapes sorted into the sets 𝑨 and 𝑩. 𝑨 = {triangles} and 𝑩 = {blue shapes}. Show the region for the complement of set 𝑩, 𝑩’.
Image caption,
Set 𝑨 contains all the triangles. The elements of set 𝑨 are one purple, one black and one blue triangle. Set 𝑩 contains all blue shapes. The elements of set 𝑩 are a blue trapezium, a blue square, a blue triangle and a blue pentagon.
Image caption,
The complement of set 𝑩, 𝑩’, contains all the elements that are not in set 𝑩. These shapes are in the region outside the circle that represents set 𝑩. The elements in 𝑩’ are a purple triangle, a purple rectangle, a black triangle, a black circle and a purple hexagon. The region is outside the circle that represents set 𝑩.
Image caption,
ξ = {factors of 100}. 𝑨 = {factors of 20}. 𝑩 = {multiples of 10}. Identify and describe the region of 𝑨’, the complement of set 𝑨, on a Venn diagram.
Image caption,
List the elements of each set ξ = {1, 2, 4, 5, 10, 20, 25, 50, 100}. 𝑨 = {1, 2, 4, 5, 10, 20} and 𝑩 = {10, 20, 50, 100}. The Venn diagram shows the sets. The complement of set 𝑨, 𝑨’, contains all the elements that are not in set 𝑨.
Image caption,
𝑨’ is the complement of set 𝑨, containing all the elements not in set 𝑨. These elements are outside the circle of set 𝑨. 𝑨’ may be described as ‘the factors of 100 that are not factors of 20’.