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Set Theory

Set Theory

MODULES

Set Notations
Types of Sets
Set Operations & Venn Diagram
2 Set Venn Diagrams
3 Set Venn Diagrams
4 Set Venn Diagrams
Maximum and Minimum
Past Questions

CONCEPTS & CHEATSHEET

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Set Theory - 1
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PRACTICE

Set Theory : Level 1
Set Theory : Level 2
Set Theory : Level 3
ALL MODULES

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\phiϕ A === {}, B === ϕ\phiϕ
Singleton set Set with exactly 111 element A === {000} , B === {111}, C === {594859485948}
Finite set Number of elements is countable or finite A === {1,3,51, 3, 51,3,5} , B === {2,2,4,5,52, 2, 4, 5, 52,2,4,5,5}
Infinite set Set with infinite number of elements A === {1,2,3,...1, 2, 3, ...1,2,3,...}, B === {4,7,10,13,...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, 62,4,6}, B === {4,2,64, 2, 64,2,6}, C === {2,6,4,6,4,22, 6, 4, 6, 4, 22,6,4,6,4,2}, D === {2,42, 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, 31,2,3,3}
B === {1,2,3,4,51, 2, 3, 4, 51,2,3,4,5}
C === {1,2,3,4,5,61, 2, 3, 4, 5, 61,2,3,4,5,6}
D === {1,2,3,4,51, 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 BA=B, then A⊂BA \subset BA⊂B.
In the above example, the set B is a proper subset of set C, as the element '666' 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, 20,1,2}
P(A) === {{}, {000}, {111}, {222}, {0,10, 10,1}, {1,21, 21,2}, {2,02, 02,0}, {0,1,20, 1, 20,1,2}}

If there are n distinct elements in a set, then its power set will have 2n2^{\text{n}}2n 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}}2n. 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, 41,2,3,4}, B === {2,3,4,52, 3, 4, 52,3,4,5}, C === {1,8,91, 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, 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}Ac. This is covered in greater detail in the subsequent sections.
U === {1,2,3,4,51, 2, 3, 4, 51,2,3,4,5}, A === {2,32, 32,3}, then A' === {1,4,51, 4, 51,4,5}

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