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Sets are essentially a collection of different items that constitute a group in mathematics. A set can have many elements, like numbers, days of the week, car types, and many more examples. The element of the set refers to each object present in the set. When we write a set then we use curly brackets. As an example of a set, consider the following set A = {1,2,3,4,5} . A set having different items can be represented using the various notations.
A set is the well-defined collection of things/objects in mathematics. Generally the capital letters are used to denote the name and to represent sets.
A collection of even natural numbers smaller than 10 is defined, but a class of clever pupils is not. The set A = {1, 3, 5, 7, 9} can be used to represent a collection of odd natural numbers less than 10.
Element of a set
The items in a set are referred to as elements or members of the set. Curly brackets separate the elements in a set, which are separated by commas. The sign “∈” is used to indicate that an element is part of a set. In the preceding example, 2 ∈ A. The symbol “∉” is used to represent an element that is not a member of a set. In this case, 3 ∉ A.
Cardinal Number of a set
The total number of items in a set is denoted by the cardinal number, cardinality, or order of the set. N(A) = 4 for natural even numbers smaller than 10.
One key requirement for defining a set is that all of its components be related to one another and share a similar feature. For example, if the elements of a set are the names of months in a year, we can say that all of the set’s elements are the months of the year.
Symbols in set
There are various types of symbols that are used in the set are given below :-
Symbol | Symbol Name | Meaning |
{ } | set | The collection of elements in a set |
A ∪ B | union | Elements that belong to set A or set B |
A ∩ B | intersection | Elements that belong to both the sets, A and B |
A ⊆ B | subset | The subset A has few or all elements equal that is equal to the set B |
A ⊄ B | not subset | The set A is not the subset of the set B |
A ⊂ B | proper subset or strict subset | subset A has less elements than that of set B |
A ⊃ B | proper superset or strict superset | The set A has more elements than that of the set B |
A ⊇ B | Superset | The set A has more elements or that of the same elements as that of the set B |
Ø | empty set | Ø = { } |
P (C) | power set | all subsets of C |
A ⊅ B | not superset | set X is not a superset of set Y |
A = B | equality | both sets have the same members |
A \ B | The relative complement | objects that belong to A but does not belongs to B |
Ac | complement | all the objects that do not belong to set A |
A ∆ B | The symmetric difference of the set | objects that belong to A or B but not belongs to their intersection |
a∈B | element of | set membership |
(a,b) | ordered pair | collection of 2 elements |
x∉A | not element of | no set membership |
|B|, B | Cardinality of the set | the total number of elements of the set B |
A×B | cartesian product of two sets | The set of all ordered pairs from A and B |
N1 | natural numbers / whole numbers set (without zero) | N1 = {1, 2, 3, 4, 5,…} |
N0 | natural numbers / whole numbers set (with zero) | N0 = {0, 1, 2, 3, 4,…} |
Q | rational numbers set | Q= {x | x=a/b, a, b∈Z} |
Z | integer numbers set | Z= {…-3, -2, -1, 0, 1, 2, 3,…} |
C | complex numbers set | C= {z | z=a+bi, -∞<a<∞, -∞<b<∞} |
R | real numbers set | Set of all numbers |
Properties of set
There are various properties that is followed by a set are given below
Property | Example |
Commutative Property | A U B = B U A A ∩ B = B ∩ A |
Associative Property | (A ∩ B) ∩ C = A ∩ (B ∩ C) (A U B) U C = A U (B U C) |
Distributive Property | A U (B ∩ C) = (A U B) ∩ (A U C) A ∩ (B U C) = (A ∩ B) U (A ∩ C) |
Identity Property | A U ∅ = A A ∩ U = A |
Complement Property | A U A’ = U |
Idempotent Property | A ∩ A = A A U A = A |
Conclusion
Set theory is a mathematical concept that was created to explain groupings of items. “It is a collection of elements,” according to the definition. Numbers, alphabets, variables, and other items could be included. The intersection of sets, the union of sets, the difference of sets, and other operations on sets are represented by notation and symbols.
