Prove that the relation R in set of integers Z given by R = {(a, b) : (a - b) is divisible by number 5} is an equivalence relation.

**OR**

If f : R → R and g : R → R are defined as f(x) = cos x and g(x) = 3x^2, then find gof and fog. Prove that gof ≠ fog.

Question

Prove that the relation R in set of integers Z given by R = {(a, b) : (a - b) is divisible by number 5} is an equivalence relation.

**OR**

If f : R → R and g : R → R are defined as f(x) = cos x and g(x) = 3x^2, then find gof and fog. Prove that gof ≠ fog.

Accepted Answer

>

> [!TLDR]
> Check the three defining properties — reflexivity, symmetry, transitivity — directly from the divisibility condition $(a-b)$ divisible by 5.
>
> Given: $R = \{(a,b) : a,b \in \mathbb Z,\ 5 \mid (a-b)\}$.
>
> **Reflexive:** For any $a \in \mathbb Z$, $a-a=0$, and $5\mid 0$. So $(a,a)\in R$ for all $a$ — $R$ is reflexive.
>
> **Symmetric:** Suppose $(a,b)\in R$, i.e. $a-b=5k$ for some integer $k$. Then $b-a = -5k = 5(-k)$, which is also divisible by 5. So $(b,a)\in R$ — $R$ is symmetric.
> …

Q.Prove that the relation R in set of integers Z given by R = {(a, b) : (a - b) is divisible by number 5} is an equivalence relation. OR If f : R → R and g : R → R are defined as f(x) = cos x and g(x) = 3x^2, then find gof and fog. Prove that gof ≠ fog.

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Q.Prove that the relation R in set of integers Z given by R = {(a, b) : (a - b) is divisible by number 5} is an equivalence relation. OR If f : R → R and g : R → R are defined as f(x) = cos x and g(x) = 3x^2, then find gof and fog. Prove that gof ≠ fog.

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