Reflexive
Relation
It
is a binary element in which every element is related to itself.
Let A be a set and R be the
relation defined on it.
R is set to be reflexive, if (p, p) ∈ R for
all p ∈ A that
is, every element of A is R-related to itself, in other words pRp for every p ∈ A.
A relation R on a set A is not
reflexive if there is at least one element p ∈ A such that (p, p) ∉ R.
Consider, for example, a set A =
{l,m,n,o}.
The
relation R
But
the relation R
Solved example of reflexive relation on set:
1. A
relation R is defined on the set Z (set of all integers) by “aRb if and only if
2a + 3b is divisible by 5”, for all a, b ∈ Z. Examine if R is a reflexive relation on Z.
Solution:
Let b ∈ Z. Now 2b + 3b = 5b, which is divisible by
5. Therefore bRb holds for all b in Z i.e. R is reflexive.
2. A
relation R is defined on the set Z by “aRb if a – b is divisible by 5” for a, b
∈ Z.
Examine if R is a reflexive relation on Z.
Solution:
Let p ∈ Z. Then p – p is divisible by 5. Therefore pRp holds for all p in Z i.e. R is
reflexive.
3. Consider
the set Z in which a relation R is defined by ‘aRb if and only if a + 3b is
divisible by 4, for a, b ∈ Z. Show
that R is a reflexive relation on on setZ.
Solution:
Let a ∈ Z. Now a + 3a = 4a, which is divisible by 4.
Therefore aRa holds for all a in Z i.e. R is reflexive.
4. A
relation ρ is defined on the set of all real numbers R by ‘xρy’ if and only if |x
– y| ≤ y, for x, y ∈ R. Show
that the ρ is not reflexive relation.
Solution:
The relation ρ is not reflexive
as x = -2 ∈ R but |x
– x| = 0 which is not less than
-2(= x).
Very good!!
ReplyDeleteexplained well!!!
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