Proper Subset - Definition
#ALG-2-9
Category
Description
An introduction to proper subsets.
Course(s)
Mini-Lesson
085
x
Algebra
091
x
Definition(s)
093
x
Sub-category
095
x
097
1.2
Set Theory
100
x
ADN
E18
x
PHT
x
Sets
A set is a collection of objects, which are called elements or members of
the set. A set is well-defined if its contents can be clearly determined. In
other words, the contents can be clearly named or listed based on fact
and not opinion. Sets are generally named with capital letters.
Subset
Set A is a subset of set B, symbolized by A ⊆ B, if and only if all the
elements of the set A are also elements of set B. The symbol ⊈ is used to
indicate “is not a subset”.
(The phrase “if and only if” means the sentence can be read in either
direction:
- If A ⊆ B, then all the elements of the set A are also elements of set B
- If all the elements of the set A are also elements of set B, then A ⊆ B.)
Proper Subset
Set A is a proper subset of set B, symbolized by A ⊂ B, if and only if all the
elements of the set A are also elements of set B, and set A ≠ set B.
(Set B must contain at least one element not in set A.)
Rule
Example
Familiarize yourself with proper subsets.
For the following sets, determine:
𝑨 ⊂ 𝑩, 𝑨 ⊆ 𝑩, 𝑩 ⊂ 𝑨, 𝑩 ⊆ 𝑨, 𝒐𝒓 𝒏𝒐𝒏𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒃𝒐𝒗𝒆.
𝑨 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓}
𝑩 = {𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗}
𝑨 ⊂ 𝑩 𝒂𝒏𝒅 𝑨 ⊆ 𝑩
1, 2, 3, 4, 5 are all of the elements of set A.
1, 2, 3, 4, 5 are also elements of set B.
Set B contains at least one element not in set A.
ALG-2-9 | Page 1
Remember!
A set is a subset if ALL of the elements are contained in another set. In
order to be a proper subset, one set must be larger than the other.
Practice
Problems
For the following sets, determine:
𝐴 ⊂ 𝐵, 𝐴 ⊆ 𝐵, 𝐵 ⊂ 𝐴, 𝐵 ⊆ 𝐴, 𝑜𝑟 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒.
1. 𝐴 = {𝑎, 𝑏, 𝑐}
𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒}
2. 𝐴 = {1, 2, 3, 4, 5}
𝐵 = {1, 2, 3, 4, 5}
3. 𝐴 = {𝑎, 𝑏, 𝑐, 1, 2, 3}
𝐵 = {1, 2, 3, 𝑥, 𝑦, 𝑧}
See also
ALG-2-1: set - definitions
ALG-2-9: subset - definition
Answers
𝐴 ⊂ 𝐵 𝑎𝑛𝑑 𝐴 ⊆ 𝐵
Practice
𝐴 ⊆ 𝐵 𝑎𝑛𝑑 𝐵 ⊆ 𝐴
𝑏𝑢𝑡
𝐴 ⊄ 𝐵 𝑎𝑛𝑑 𝐵 ⊄ 𝐴
ALG-2-9 | Page 2
𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
𝐴 ⊄ 𝐵 ,𝐴 ⊈ 𝐵 ,
𝐵 ⊄ 𝐴, 𝐵 ⊈ 𝐴