Sets
In daily life, we have to deal with collections of objects.
For example: (i) collection of odd natural numbers less than 15 (ii) all vowels of English alphabet etc.
Each of collection is a set
We will discuss here about the different properties of sets
Relations
The concept of relation refers to association of two objects.
Let A, B be two (non-empty) sets then a relation from A to B is a rule which associates elements of sets A to elements of set B.
We shall discuss different kind of relation
Representation of Set , Types of Sets and Word Problems on Sets
Representation of a set can be in following 3 ways
- Description set
- Roaster method or table method
- Set builder method
- Finite set
- Infinite set
- Empty set
- Universal set
- Cardinal number of a finite set
- Equal sets
- Equivalent sets
- Overlapping sets
- Disjoint sets
10. Subsets
11. Proper subsets
Word problems consists of a condition in which equation are formed using basic operation on sets
Basic Operations on Sets and Venn Diagrams
Different kind of sets has an algebra in which certain kind of operation are applied these operations are
- Union of sets
- Intersection of sets
- Difference of sets
- Complement of sets
Often diagrams help us to understand and solve problems. Many ideas about sets and various relationships between them can be visualized by means of geometric figures known as Venn diagrams. Usually the universal set ? is represented by a rectangle and its subsets by closed figures with in the rectangle, inside the diagram. Sometimes points are not marked, only the elements are written inside the diagram.
Different Kind of Relations, Principe of Inclusion and Exclusion
- Reflexive relation
- Symmetric relation
- Transitive relation
- Anti symmetric relation
Principle of Inclusion and Exclusion
Let A and B to non empty sets then
N (A ? B) = N (A) + N (B) - N (A ? B)
Where N (A ? B) = number of elements in (A ? B)
N (A) = number of elements in A
N (B) = number of elements in B
N (A ? B) = number of elements in A
N (A) = number of elements in A
N (B) = number of elements in B
It is called principle of Inclusion and Exclusion
Cartesian Product of Two Sets and Binary Relation
Cartesian product of two sets: The Cartesian product of two sets, X and Y, is the set of all ordered pairswhose first member is an element of set X, and whose second member is an element of set Y. The Cartesian product is written as X × Y. Cartesian product is also known as direct product. The concept of Cartesian product was named after Rene Descartes.
X × Y = {(x, y) | x €X and y € Y}
Binary Relation: Suppose A is a non-empty set. A relation R in a set A is the subset of A × A. Clearly both the co-ordinates of the ordered pairs in R are the elements of the set A. This relation R in the set A called a binary relation.
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