Basics of Relations

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.

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.

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.

3 1 4 2 5 B A

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”.

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)}

Properties of Relations Reflexive Relation Symmetric Relation Transitive Relation

A relation R on set A is said to be reflexive on A if (a,a) ∈ R i.e aRa ∀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

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

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.

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.