Set
A set is defined as a well-defined, unordered collection of distinct objects, known as the elements or members of the set, of the same type or class of objects. An object can be numbers, alphabets, names etc.
Examples of sets are:
- A set of alphabets.
- A set of even numbers between 1 and 100.
Generally sets are denoted by the capital letters A, B, C etc., while the elements/fundamentals of the set by lowercase letters a, b, c etc.
If A is a set and a is one of the element of A, then we denote it as a ∈ A.
Here, the symbol ∈ means - "Element of or Belong to".
The statement that a is not an element of A or a does not belong to A, is denoted as a ∉ A.
Principle of Extension
A set is completely determined when its all members are specified. This is formally known as Principle of Extension. Thus two sets A and B are equal if and only if they have the same members.
Principle of Abstraction
Describing a set in terms of a property is known as the Principle of Abstraction. Consider a set A which has some elements having a same property.
Standard Notations
| x ∈ A | x belongs to A or x is an element of set A |
| x ∉ A | x does not belong to set A or x is not an element of set A |
| ∅ | Empty Set |
| U | Universal Set |
| N | The set of all natural numbers. {1, 2 ,3 ,4} |
| I | The set of all integers. {-3, -2, -1, 0, 1, 2, 3} |
| I0 | The set of all non-zero integers. {-3, -2, -1, 1, 2, 3} |
| I+ | The set of all positive integers. {1, 2 ,3, 4} |
| Q, Q0, Q+ | The set of all rational, non-zero rational, positive rational numbers respectively. {x : p/q, q ≠ 0} |
| R, R0, R+ | The set of all real, non-zero real, positive real numbers respectively. {-1, -0.3, 0, 2, 1.6, √5} |
| C, C0 | The set of all complex, non-zero complex numbers respectively. {x : a + ib, a, b ∈ R} |
Cardinality of a set
Cardinality of a set is defined as the number of unique elements in the set.
For a set A, it is denoted as |A|, #A, card (A) or n (A), where n is the "cardinality of the set".
Representation of Sets
Sets are represented in two forms:
- Roster or Tabular form: In this form of representation, we list all the elements of the set within curly brackets or braces {} and separate them by commas. Used with simple, small sets.
- Set builder form: In this form of representation, we list the properties fulfilled by all the elements of the set. Used with complex analysis, large sets.
Example:
If A = set of all positive even numbers less than 11, then
Roster form of set A is
A = {2, 4 ,6 ,8, 10}
Set builder form of set A is
A = {x : 2n, n ∈ N, 1 ≤ n ≤ 5}
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