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Basic Set Theory

 

A set is a well-defined collection of data, objects or living beings. It is a mathematical logic to represent a collection of similar data. The objects in a set are called elements or members. For example, if we want to denote the cricket players of India, then we need to make one set. If we need to represent players of Australia, then we need to have another set. The combination of all the players will form a universe of sets.

Types of sets

 1. Finite set: Finite number of elements are present in the set.
 2. Infinite set: Infinite number of elements are present.
 3. Empty set: No elements are present.
 4. Singleton set: Only one element is present.
 5. Equal set: If the elements are equal, then sets are equal.
 6. Equivalent set: In this, the same number of elements are present.
 7. Power set: A set of every possible subset.
 8. Universal set: It contains all sets.
 9. Subset: When all the elements of set A belong to set B, then A is a subset of B.

Set notations

Symbol Symbol Name
{ } set
A ∪ B A union B
A ∩ B A intersection B
A ⊆ B A is subset of B
A ⊄ B A is not subset B
A ⊂ B proper subset / strict subset
A ⊃ B proper superset / strict superset
A ⊇ B superset
A ⊅ B not superset
Ø empty set
P (C) power set
A = B Equal set
Ac Complement of A
A ∈ B A element of B
X ∉ A X not element of A

Formulas

  • n ( A ∪ B ) = n (A) + n (B) – n (A ∪ B)
  • n (A ∪ B) = n (A) + n (B) {when A and B are disjoint sets}
  • n (U) = n (A) + n (B) – n (A ∩ B) + n ((A ∪ B)꜀)
  • n (A ∪ B) = n (A − B) + n (B − A) + n (A ∩ B)
  • n (A − B) = n (A ∩ B) − n (B)
  • n (A − B) = n (A) – n (A ∩ B)
  • n (A꜀) = n (U) – n (A)
  • n (P ∪ Q ∪ R) = n (P) + n (Q) + n (R) – n (P ⋂ Q) – n (Q ⋂ R) – n (R ⋂ P) + n (P ⋂ Q ⋂ R)

Properties of set operations

 1) Set obeys the properties such as associativity and commutativity.
 2) The intersection of sets is distributed over the union of sets.
 3) Sets are used to define functions in mathematics.
 4) Sets help to formulate, interpret and analyse the data to find a particular solution of the given condition.

Example: Give a few examples of finite sets.

Solution:

A few examples of finite sets are:

Number of days in a week

Number of months in a year

Number of minutes per hour

Natural numbers from 0 to 100.

Note: Number of days in a year will not be a finite set as a leap year has an additional day every four years. This makes the data non-finite.

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