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Transitive, Symmetric, Reflexive and Equivalence Relations *March 20, 2007*

*Posted by Ninja Clement in Philosophy.*

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**Transitivity**

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.

A relation R is *non-transitive* iff it is neither transitive nor intransitive. For example, *likes* is a non-transitive relation: if John likes Bill, and Bill likes Fred, there is no logical consequence concerning John liking Fred.

**Symmetricity**

A relation R is *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 a*symmetric* 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.

**Reflexivity**

A relation R 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.

**Equivalence**

A relation R is an *equivalence* iff R is transitive, symmetric and reflexive. For example, *identical* is an equivalence relation: if x is identical to y, and y is identical to z, then x is identical to z; if x is identical to y then y is identical to x; and x is identical to x.

Reference: The Philosophy Dept. Vade Mecum: A Survival Guide for Philosophy Students, by Darren Brierton.

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I learned this topics so before but you are the only one who explained it clearly. thank you.

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Thanks dude helped me a lot.

Thank you a lot. Its a great help to me. Ahh. could you also give a definition of what transitivity, symmetricity, reflexivity are? Thanks

pls, i have not undersood the concept of antisymmetric. Can u please bail me out with counter example if there is any?

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Just go on…;)

so are very difficult this subject

How can we get the no. of equivalent relation in a given set?

Excellent explanation, if u had put some examples that would be much helpful, helped me a lot thanks.

Thanks for giving me a actual definition with so exact and easy example

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That’s a great piece of explanation.I got the real idea of symmetric and other relations by the excellent examples given by you.I was cleared upon that points only after reading this explanations.Than you very much!

clearly explained

Hi.You know the way a relation is transitive if you have a set A and (a,b),(b,c) and (a,c) .What happens if in set A there are more than 3 elements a,b,c and we have a,b,c and d.How do I aply this rule to find out if A={a,b,c,d} is transitive.Thanks a lot

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fantastic! I only wish you included a good explanation for Antisymmetric!

I would rather say.. If you would have explained it with the mathematical equation. But no worry I found complete tutorial on

Forgot to mention the url

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Wow! That was a great way to explain the real concept. Children nowadays enforce just on solving equation, and no one worries about the logic behind.

Teachers too are getting the same.

But! You bravo! We all need such a teacher!

Hey, but please! make that clear what if DOMAINS & CO-DOMAINS are not the same Set.

Hello Ninja,

I need your help to solve the following problem :

Let F be a function on the integer given by f(n) = sqr(n-2). a) show that the relation R = { (x,y) are integers nad f(x) = f(y) is reflexive, symmetric and transitive relation.

b) Describe the partition of the integers induced by R.

Thanks you

thanks a lot. now i got what these properties of relation.i have a concept about these now…..bless you

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greetings from Bulgaria

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thanks

=)

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I want to know what’s the answer is,

(a,b) ~ (c,d) if a+d=b+c

which of following is/are correct

A. ~ is symmetric

B. ~ is an equivalence relation

C. ~ is transitive

D. ~ is reflexive

E. ~ is not an equivalence relation.

THANK YOU VERY MUCH!AM DONE!PLEASE CONTINUE HELPING US!

Thank God for the examples, I’m clear now. Writing an exams on it tomorrow.

fantastic! I only wish you included a good explanation for reflexive

I want some logical explanation with good example of reflexive relation !!!

very clear explanations in every property of relation.. so easy to understand.

thanks alot!!

Thankyou… :)

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your explanation is really simple and easy to understand. so, please post in other topic as well.. thanks

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wow, you explain it so clear, theanks!, but where is the anti-symmetric? is it same with non-symmetric?

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@ Sammy,

But need some examples

Not geting clearly

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d explanation is detailed n clear, thanx we can conque wit u.

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MY SEMINAR

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thank you for such simple and very understandable exaples… :D

When I initially commented I seem to have clicked the -Notify me

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every time a comment is added I receive four emails with

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Cheers!

God bless you with his kindness

Nice!

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X is a wife of y? is it transitive relation?

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please paste one easy and one hard examples for each relation.

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Thanks alots this explanation on Refleive,Symmetric and Transitive relations help me to undertand a relation with regard to a real life situation,not just only on sets.

i think m now cristal clear… but not about anty symmetry

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Relation R is a equivalance relation iff R is reflexible , symmetirc and transitive relation .