Prove or disprove the relation is equivalence

Question

Define the relation $R$ on $ \mathbb{N}\ast \mathbb{N} ​ $ by $(a,b)R(c,d)$ if $a+2b = c+2d$. Prove or disprove this as an equivalence relation.

Do we prove this by showing that the relation has properties of symmetry , reflection and transitivity

Answer 1

Here it makes sense to proof a more general statement (first because the proof becomes more clear by not being cluttered with irrelevant details and second because it will help you build an intuition for equivalence relations).

proof that every relation of the form

$xRy \Leftrightarrow f(x) = f(y)$

for a given function $f$ is an equivalence relation (the converse is also true).