Show that the relation R on the set Z of integers, given by R={(a,b):2 divides a−b} is an equivalence relation.

Question

Show that the relation R on the set Z of integers, given by R={(a,b):2 divides a−b} is an equivalence relation.

Filo · Accepted Answer

The relation R on Z is given by R={(a,b):2dividesa−b}We observe the following properties of relation RReflexivity: for any a∈Za−a=0=0×2⇒ 2 divides a−a⇒(a,a)∈RSo, R is relexive relation on ZSymmetry: Let a,b∈Z be such that(a,b)∈R⇒ 2 divides a−b⇒ a−b=2λ for some λ∈Z⇒ b−a=2(−λ), where −λ∈Z⇒ 2 divides b−a⇒(b,a)∈RThus (a,b)∈R⇒(b,a)∈R. So, R is a symmetric relation on Z.Transitivity. Let a,b,c∈Z be such that (a,b)∈R and (b,c)∈R. Then,(a,b)∈R⇒2 divides a−b⇒a−b=2λ for some λ∈Zand (b,c)∈R⇒2 divides b−c⇒b−c=2μ for some μ∈Z∴ a−b+b−c=2(λ+μ)⇒ a−c=2(λ+μ), where λ+μ∈Z⇒ 2 divides a−c⇒ (a,c)∈Rthus, (a,b)∈R and (b,c)∈R⇒(a,c)∈RSo, R is a transitive relation on ZHence, R is an equivalence relation on Z

Question Text	Show that the relation $R$ on the set $Z$ of integers, given by $R = {(a, b) : 2 divides a - b}$ is an equivalence relation.
Answer Type	Text solution:1
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Show that the relation $R$ on the set $Z$ of integers, given by $R = {(a, b) : 2 divides a - b}$ is an equivalence relation.

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