1 / 12

Basics of Relations

Explore the basic elements of relations in mathematics through set theory and algebra, with examples illustrating reflexive, symmetric, and transitive properties.

hermanjones
Download Presentation

Basics of Relations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Basics of Relations

  2. Introduction • Human Language has many words and phrases to describe the relationship between or among objects. • It may be that for two people A and B, that A is parent of B, A is an ancestor of B, A is taller than B. • In algebra it may be that value of variable x is less than the value of variable y. • In set theory, it may be that a set X is subset of set Y or x is disjoint from Y. • All above notions are special instances of relation.

  3. Definition • Let A and B are two sets. • Then, a subset of AXB is called a Relation or Binary relation from A to B. • Thus, if R is a relation from A to B, then R is set of ordered pairs (a,b) where a∈A and b∈B. • If (a,b)∈ R, we say that “ a is related to b by R”. This is denoted as aRb.

  4. Example • Consider two sets A={0,1,2}, B={3,4,5}. • Let R={(1,3),(2,4),(2,5)}. • Evidently R is a subset of AXB. • So R is a relation from A to B. • 1R3, 2R4,2R5. • This can be depicted in a diagram called Arrow diagram.

  5. 3 0 1 4 2 5 B A

  6. Example2: • Consider sets A={0,1,-1} and B={2,-2}. • Let R1={(0,2),(1,2),(-1,2)} and • R2={(0,-2),(1,-2),(-1,-2)} • Clearly both R1 and R2 are subsets of AXB and therefore relations from A to B. • We observe that R1 consists of elements(a,b)∈ AXB for which the relationship a<b holds. • Hence, here aR1b is read as “ a is less than b” • The symbol R1 is stands for the phrase “is less than”. • Similarly R2 consists of elements(a,b) ∈ AXB for which the relationship a>b holds. • The symbol R1 is stands for the phrase “is greater than”.

  7. Inverse of a Relation • Let R be a relation from A to B. Then the inverse of the relation R from B to A is denoted as R-1 and defined as • R-1={(b,a)|(a,b)∈ R} • Ex: if R={(2,4),(2,6),(3,6)} then • R-1={(4,2),(6,2),(3,6)}

  8. Properties of Relations • Reflexive Relation • Symmetric Relation • Transitive Relation

  9. A relation R on set A is said to be • reflexive on A if (a,a)∈ R i.eaRa ∀a ∈ A. • Symmetric on A if (a,b) ∈ R then(b,a) ∈ R for a,b ∈ A • Tranasitive on A if (a,b) ∈ R, (b,c) ∈ R the (a,c) ∈ R for a,b,c ∈ A

  10. Compatibility Relation • A Relation R on set A which is both reflexive and symmetric is called Compatibility relation on A.

  11. Antisymmetric Relation • A relation R on a set A is said to be antisymmetric if whenever (a,b) ∈ R and (b,a)∈ R then a=b.

  12. Equivalence Relations A Relation R on set A is said to be an equivalence relation if • R is reflexive • R is symmetric • R is Transitive on A. Every equivalence relation is a compatibility relation as well.

More Related