Determining whether a relation is an equivalence relation

Question

The question asks:

Define a relation $R$ on the set of functions from $R$ to $R$ as follows:

$$(f,g) \in R \text{ if and only if } f(x) − g(x) \geq 0 \text{ for all } x \in R$$

Is this relation reflexive? Symmetric? Transitive? Is it an equivalence relation? Explain.

So far I have that the relations is reflexive because $f(x)-f(x) \geq 0$, which is true.

But I'm not quite sure if the relation is symmetric or transitive as I am not quite familiar.

Answer 1

Reflexive $$f(x)-f(x)\geq 0 \forall x\in \mathbb{R}$$ Yes it is reflexive.

Transitive $$f(x)-g(x)\geq 0 \forall x\in \mathbb{R}$$ $$g(x)-h(x)\geq 0 \forall x\in \mathbb{R}$$ Add above equations, $$\Longrightarrow f(x)-h(x)\geq 0 \forall x\in \mathbb{R}$$ Yes it is transitive.

Symmetric

$$f(x)-g(x)\geq 0 \forall x\in \mathbb{R}$$ $$g(x)-f(x)\leq 0 \forall x\in \mathbb{R}$$ Hence, $(f,g)\in R $ & $ (g,f)\in R$ iff $g=f$ Hence, this relation is not symmetric.

Hence, not equivalence relation.

Hope it helps:)