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Basics of Relations

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.

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Basics of Relations

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  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.

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