CBSE Class 12-science Answered
if Z is the set of all integers and R is the relation on Z defined as R ={(a, b) :a b € z and a-b is divisible by 5 } prove that x is an equivalence relation
Asked by aiswaryaanil0009 | 01 Feb, 2024, 08:36: PM
Expert Answer
Relation R is given as R ={(a, b) :a b € z and a-b is divisible by 5 }
Reflexive:
For any integer a, we have a - a = 0, which is divisible by 5.
So, aRa.
Symmetric:
Let aRb i.e. (a - b) is divisible by 5.
i.e. a - b = 5k for some integer k
Therefore, b - a = 5m for some integer m = -k
So, (b - a) is divisible by 5.
Therefore, bRa.
Transitive:
Let aRb and bRc
Therefore, (a - b) and (b - c) are divisible by 5.
Therefore, (a - b) = 5k & (b - c) = 5m for integers k & m.
Now, (a - c) = (a - b) + (b - c) = 5k - 5m = 5(k - m).
Here, (k - m) is also an integer.
So, (a - c) is divisible by 5.
Therefore, aRc.
So, R is reflexive, symmetric and transitive.
Hence, T is an equivalence relation.
Answered by Renu Varma | 02 Feb, 2024, 11:15: AM
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