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

Posted by Ninja Clement in Philosophy.


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.


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 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.


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.


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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1. ajinkya - May 13, 2008

excellent explaination thanks 2 ths info i can now get my score more by min 12 marks. i owe u my bright future

2. Ravimal - August 23, 2008

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

3. rudy - November 21, 2008

Thanks a lot, cause I use this info to complete my course work

4. Razor - December 15, 2008

Thanks dude helped me a lot.

5. Rosver - January 13, 2009

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

adebiyi demola - February 10, 2013

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

6. David - March 4, 2009

Excellent explanation, helped me a lot thanks

7. tarantullas - June 9, 2009

Thanks dear friend, it helped me a lot.
Just go on…;)

8. dada - July 3, 2009

so are very difficult this subject

9. Mukul - July 12, 2009

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

10. saim pasha - September 13, 2009

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

sandeep chauhan - January 29, 2014

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

11. moriz - September 13, 2009

You’re great! two thumbs up..

12. chirag - September 21, 2009

hey really good explanation

13. NANA - October 28, 2009

thank you very much.It was really helpful!

14. nimesha devinda - October 28, 2009

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!

15. dumiani ndlovu - October 30, 2009

clearly explained

16. Daniel Coman - November 12, 2009

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

17. Sharad - November 23, 2009

thanks its really very helpful

18. Satishkumar - December 7, 2009

Very helpful… Thnx :)

19. NANKOKONUMBI M. - January 3, 2010

a little bit fine,but go more deep!

20. NANKOKONUMBI M. - January 3, 2010

good question boy,the same thing makes me headache!any soln found yet?

21. john r - January 21, 2010

this info better help i am reading it now

22. Neukr - February 7, 2010

thank u!! it really helps!

Hi from Bulgaria :)

23. surendra - March 2, 2010

wonderful ……thank you ….you helped me a lot

24. Zurksy - March 17, 2010

Thanks alot…..love u ;)

25. qwey - April 15, 2010

thanks a lot

i understood the concept clearly

wonderful explanation…………………………………

26. DiscreteMath_4ever - April 18, 2010

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

27. Relation_bash - August 5, 2010

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

28. Relation_bash - August 5, 2010

Forgot to mention the url


29. Izharul Haque Aazmi - October 8, 2010

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.

30. Fitrah - October 11, 2010

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

31. SAQIB - October 27, 2010

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

32. mohit - January 31, 2011

woooooooh……i wasted my 2 hours fo this….
awesome xplanation…

33. rupesh - February 2, 2011

superb explanation….
i understood very easilyyy.

34. shayak c - April 18, 2011

liked ur site. good lively explanations.concepts r now wel cleared. hope 2 get such help in future….

35. Oleg - June 4, 2011

Good stuff man,
greetings from Bulgaria

36. jannifar - June 23, 2011

thanx for this.it give realy help in my study………………..

37. Dinah - June 24, 2011

Thanks! the explanation is very simple…

38. sajroxs - June 27, 2011

wow! nice explan. juest from this article i understood this topics

39. itsmesimplydinah - June 28, 2011


40. daoud - July 25, 2011

Thanks to the infinity, the topics help me a lot

41. Dilhani - August 8, 2011

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.

42. JAAAFA - August 8, 2011


43. Agatha Ukari - August 16, 2011

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

44. gaurav - August 31, 2011

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

45. gaurav - August 31, 2011

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

46. terlyn - September 8, 2011

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

47. Saad Saeed - September 26, 2011

thanks alot!!

48. Muhammad Ali Shahzad - October 4, 2011

Thankyou… :)

49. Anil Nembang - October 27, 2011

Really really excellent…you explanation is really simple and easy to understand. so, please post in other topic as well.. thanks

50. Anil Nembang - October 27, 2011

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

51. Suraj Kumar Kukreja - November 8, 2011


52. Erinfolami timothy - December 27, 2011

I love dis site it has really helped me.kudos to you guyz

53. alok singh - December 27, 2011

thanks theas consept is very clear i naver forget theas consept

54. Sontosh - February 19, 2012

Very easily described

55. Claudio - March 13, 2012

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

56. shams sarfraz noori - May 15, 2012

the concept is discussed in brilliant way ….really i was totally confused …..but now i m not confuse ..thanks ……

57. Merdan Eliyev - June 15, 2012

thanks a lot for this…

58. dave - June 15, 2012

Thanks a lot, Its very clearly explained

59. Sammy - June 24, 2012

nice uploads keek it up

Basavaraj - July 9, 2012

@ Sammy,

But need some examples

60. Basavaraj - July 9, 2012

Not geting clearly

61. Anestasia - July 15, 2012

Tanks 2 u guys,keep it up.bravo!

62. tuohid University of Dhaka - July 19, 2012

now it has become more clear to me and from now i can use it in my practical life…….thanks

63. shakshi sharma - August 2, 2012

nice explanation

64. Freddy Motloutsi - August 18, 2012

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

65. RAKSHA - August 29, 2012


66. gerald michael - October 7, 2012

tanx,, t rily helped

67. Shweta - October 7, 2012

Thank you, it will really help

68. Anwar Gundam - October 17, 2012


69. edgar - November 30, 2012

thanks.its quite understandable

70. Avinav Bardalai - December 27, 2012


71. QUEEN - February 18, 2013

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

72. the paleo diet - March 6, 2013

When I initially commented I seem to have clicked the -Notify me
when new comments are added- checkbox and now
every time a comment is added I receive four emails with
the same comment. Perhaps there is a way you can remove me from that service?

73. P.khan - March 9, 2013

God bless you with his kindness

74. dongseop - March 29, 2013


75. Wiggle | funkyonmath - April 15, 2013
76. Siddharth Mishra - April 21, 2013

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

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78. Bittu kumar - May 25, 2013

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80. amad - May 31, 2013

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

81. savi - June 29, 2013

Brilliant explanation . Thanks a lot !!

82. Algorithms, Part I – Week 1 Notes (Union-Find) | stack vs heap - August 23, 2013

[…] objects, where each pair may or may not “be connected” (an equivalence relation – reflexive, symmetric, transitive). A connected component is a ‘maximal’ set of objects that are connected. Number them 0 […]

83. Eva - October 1, 2013

Thanks, And for “is in the same room” is it reflexive?

84. karolo - October 7, 2013

thanx a lot well explained

85. ManiraJ - November 7, 2013

Very nice .. Got clear.. thanks a lot..

86. Relationships | Campus Preparation - November 7, 2013
87. Ahmed - December 31, 2013

Thanks very much, this was really helpful and you made it easy to understand.

88. Ibrahim Kwaire - January 15, 2014


89. sandeep chauhan - January 29, 2014

Thanks a lot

90. Jamilu Haruna Yakubu - March 5, 2014

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.

91. biraj dhungel - April 1, 2014

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

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