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Equivalence Relation

 

A relation R on a set A is an equivalence relation if and only if

  1. R is reflexive, i.e, aRa721.pngaA
  2. R is symmetric, i.e., aRb bRa
  3. R is transitive, i.e., aRb and bRc aRc

Partial order relation :

A relation R on a set A is a partial order relation if and only if.

  1. R is reflexive, i.e. aRa 726.png aA
  2. R is antisymmetric i.e., aRb and bRa a = b
  3. R is transitive, i.e., aRb and bRc aRc.

Relation of congruence modulo m :

Let m be a fixed positive integer. Two integers a and b are said to be “congruent modulo m” if a – b is divisible by m. We write
a ≡ b (mod m)


Thus. a ≡ b (mod m) [Read as “ a is congruent to b modulo m”] 
iff a – b is divisible by m; a, bI.

Example :

  1. 25 ≡ 5 (mod 4) because 25–5 = 20 is divisible by 4.
  2. 23 ≡ 2 (mod 3) because 23 –2 = 21 is divisible by 3
  3. 20731.png3 (mod 5) because 20 – 3 = 17 is not divisible by 5

The relation “ congruence modulo m” is an equivalence relation on I.





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