plz somebody help me with equivalence relations with examples
Answers
A RELATION IS EQUIVALENCE WHEN TRANSITIVE REFLEXIVE AND SYMMETRICAL
symmetric iff, if x is related by R to y, then y is related by R to x. For example, being a cousin of is a symmetric relation: if John is a cousin of Bill, then it is a logical consequence that Bill is a cousin of John.
A relation R is asymmetric iff, if x is related by R to y, then y is not related by R to x. For example, being the father of is an asymmetric relation: if John is the father of Bill, then it is a logical consequence that Bill is not the father of John.
A relation R is non-symmetric iff it is neither symmetric nor asymmetric. For example, loves is a non-symmetric relation: if John loves Mary, then, alas, there is no logical consequence concerning Mary loving John.
is reflexive iff, everything bears R to itself. For example, being the same height as is a reflexive relation: everything is the same height as itself.
A relation R is irreflexive iff, nothing bears R to itself. For example, being taller than is an irreflexive relation: nothing is taller than itself.
A relation R is non-reflexive iff it is neither reflexive nor irreflexive. For example, loves is a non-reflexive relation: there is no logical reason to infer that somebody loves herself or does not love herself.
A relation R is transitive if and only if (henceforth abbreviated “iff”), if x is related by R to y, and y is related by R to z, then x is related by R to z. For example, being taller than is a transitive relation: if John is taller than Bill, and Bill is taller than Fred, then it is a logical consequence that John is taller than Fred.
A relation R is intransitive iff, if x is related by R to y, and y is related by R to z, then x is not related by R to z. For example, being next in line to is an intransitive relation: if John is next in line to Bill, and Bill is next in line to Fred, then it is a logical consequence that John is not next in line to Fred.
Answer:
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:
a = a (reflexive property),
if a = b then b = a (symmetric property), and
if a = b and b = c then a = c (transitive property).