Show that the relation Rin R defined by R = {(a,b): la – bl is divisible by 4} is an
equivalence relation.
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Answer:
a R b⇔a−b divisible by 3
Reflexive:
a R a⇔a−a divisible by 3..... True
Therefore, the given relation R is a reflexive relation.
Symmetric:
a R b⇔a−b divisible by 3⇒ a−b=3k, where k is an integer.
Then,
b R a⇔b−a divisible by 3
Because b−a=−(a−b)=−3k which is divisible by 3.
Therefore, the given relation R is a symmetric relation.
Transitive:
a R b⇔a−b divisible by 3⇒ a−b=3k, where k is an integer.
b R c⇔b−c divisible by 3⇒ b−c=3p, where, p is an integer
Then,
a R c⇔a−c divisible by 3
Because a−c=(a−b)+(b−c)=3k+3p=3(k+p) which is divisible by 3.
Therefore, the given relation R is a transitive relation.
Since, the given relation R satisfies the reflexive, symmetric, and transitive relation properties, Therefore, it is an equivalence relation.
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