![Equivalence Class
• Equivalence class [a] containing
a€ x for an equivalence relation R
in x is the subset of X containing
all elements b related to a.](https://image.slidesharecdn.com/presentation2vijayanpillai-151111131938-lva1-app6892/85/Presentation2-vijayan-pillai-14-320.jpg)
Presentation2 vijayan pillai
- 1. Welcome
- 2. Topic : RELATIONS PROJECT 2015 -2016
- 3. Introduction Much of Mathematics is about finding a pattern – a recognisable link between quantities that change in our daily life, we come across many pattern that characterise relation such as brother and sister, father and son, teacher and students. In mathematics also we come across many relations such as number m is less than number n, line l is parallel to line m, set a is a subset of Set B. In all these we notice that a relation involves pairs f object in certain order. In this project, we learn how to link pairs of object from two sets and then introduce relation between two object in pairs. This is a path towards the wonderful world of relations.
- 4. RELATION • The concept of the term ‘relations’ in mathematics have been drawn from the meaning of relation in English language according to which two objects or quantities are related if there is a recognisable connection or link between the two object or quantities. • Definition • A relation R from a nonempty set A to a non empty set B is the subset of cartesian product of AxB obtained by describing a relationship between the first elements of x and second elements of y of the ordered pairs in AxB. • Second element in the ordered pair is called the image of first element.
- 5. Domain The Set of all first element of the ordered pairs in a relation R from a set A to set B is called domain of the relation R. Range: The set of all second elements in a relation R from a set A to B is called the ‘range’ of the relation R. The whole set B is called condomain of the relation R. Range c condomain
- 6. REPRESENTATION OF RELATION 1)A relation may be represented algebraically either by poster method or by the set builder method. 2) An arrow diagram is visual representation of a relation.
- 7. Let A = {1,2,3,4,5,6} define a relation R from A to A by R = { 9x, y) : y = x + 1} i) Depict the relation in arrow diagram ii) Write down the domain, range, codomain of R Solution R = { (1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} For example
- 8. • Corresponding arrow diagram is 1 1 2 2 3 3 4 4 5 5 6 6 • Domain = { 1, 2, 3, 4, 5) • Range = { 2, 3, 4, 5, 6 } • Condomain = { 1,2,3,4,5,6}
- 9. Number of relations The total number of relations that can be defined from a set A to a set B is the number of possible subsets of AxB. If n(a)=p and n(b)=Q. Then n(AxB)=pq and total number of relations is 2pq.
- 10. Type of Relations • Empty Relation A relation R in a set A is called empty relation, if no element of A is related to any elements of A, ie, R = PCAxA.
- 11. Universal Relation A relation R in set a is called Universal relation, if each element of A is related to every element of A, ie, R = AxA. • Both the empty relation and the universal relations are sometimes called trivial relation.
- 12. Reflexive relation A relation R in a set A is called reflecxive, if (a,a) €R for every A€A Symmetric relation A relation R in a set A is called symmetric if (a1, a2) € R implies that (a2, a1) €R, for all a1, a2 €A Transitive Relation A relation R in a set A is called Transitive if (a1, a2) €R and (a2, a3) €R, implies that (a, a3) €r for all a1,a2, a3€A
- 13. Equivalence relation • One of the most important relation which plays a significant role in mathematics, is an equivalence relation. A relation R in a set A is said to be an equivalence relation if R is reflective, symmetric and transitive. • It can be represented as a~b • Eg. Let T be the set of all triangles in a plain with R a relation T given by • R + {(T1 T2): T1 is congruent to T2}
- 14. Equivalence Class • Equivalence class [a] containing a€ x for an equivalence relation R in x is the subset of X containing all elements b related to a.
- 15. Inversion Let R b a relation, then R-1 = {(a,b): (b,a) in R}
- 16. Thanks