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Set Theory
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CAT 2025 Lesson : Set Theory - Types of Sets

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2. Types of Sets

2.1 Basic Types

Type Characteristic Example
Null set Set with no element. Denoted by ϕ\phi A == {}, B == ϕ\phi
Singleton set Set with exactly 11 element A == {00} , B == {11}, C == {59485948}
Finite set Number of elements is countable or finite A == {1,3,51, 3, 5} , B == {2,2,4,5,52, 2, 4, 5, 5}
Infinite set Set with infinite number of elements A == {1,2,3,...1, 2, 3, ...}, B == {4,7,10,13,...4, 7, 10, 13, ...}
Equal set Where A contains every element in B and B contains every element in A, then A == BWhere A contains every element in B and B contains every element in A, then A == B Where A == {2,4,62, 4, 6}, B == {4,2,64, 2, 6}, C == {2,6,4,6,4,22, 6, 4, 6, 4, 2}, D == {2,42, 4}, we can conclude A == B == C ≠\ne D


2.2 Subset & Superset

If every element of a set A is present in set B, then A is a subset of B, and B is a superset of A. A ⊆\subseteq B denotes “A is a subset of B”, which also means B is a superset of A.
A == {1,2,3,31, 2, 3, 3}
B == {1,2,3,4,51, 2, 3, 4, 5}
C == {1,2,3,4,5,61, 2, 3, 4, 5, 6}
D == {1,2,3,4,51, 2, 3, 4, 5}

1) None of B, C or D is a subset of A
2) A and D are subsets of B -> A ⊆\subseteq B and D ⊆\subseteq B
3) A, B and D are subsets of C -> A ⊆\subseteq C, B ⊆\subseteq C and D ⊆\subseteq C
4) A and B are subsets of D -> A ⊆\subseteq D and B ⊆\subseteq D

Proper Subset: If elements of set A are contained in set B and A≠BA \ne B, then A⊂BA \subset B.
In the above example, the set B is a proper subset of set C, as the element '66' is not present in B. However, B is not a proper subset of C as B == C.

2.3 Power Set

A power set of a set, say set A, contains all distinct subsets of the set A. This is denoted by P(A).
A == {0,1,20, 1, 2}
P(A) == {{}, {00}, {11}, {22}, {0,10, 1}, {1,21, 2}, {2,02, 0}, {0,1,20, 1, 2}}

If there are n distinct elements in a set, then its power set will have 2n2^{\text{n}} elements.
[Note: As each element can either be in a subset or not, which is two possibilities, number of different combinations is 2n2^{\text{n}}. Refer to the Permutations & Combinations lesson.]

2.4 Universal Set

A set which contains all elements in a given context. This is denoted by the set U.
1) Where N is the set of natural numbers, Q is the set of rational numbers and Z is the set of integers, the universal set U could be defined as R, which is the set of all real numbers.
2) If A == {1,2,3,41, 2, 3, 4}, B == {2,3,4,52, 3, 4, 5}, C == {1,8,91, 8, 9}, then a possible universal set U == {1,2,3,4,5,6,7,8,91, 2, 3, 4, 5, 6, 7, 8, 9}

Note that the universal set that we define here is basis a context, and not in the strict sense of it being an all encompassing set.

2.5 Complement of a Set

Where the universal set U is defined, the complement of a set A contains all elements that are not in A but present in U. The complement of set A is denoted by A' or Ac \text A^{c}. This is covered in greater detail in the subsequent sections.
U == {1,2,3,4,51, 2, 3, 4, 5}, A == {2,32, 3}, then A' == {1,4,51, 4, 5}

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