So I'm pretty new to abstract mathematics being a biologist an all.
My biggest issue is that I can't really wrap my head around how to solve problems.
So I have the problem: Let $X$ be the set of all pairs $(a,b)$ where $a, b \in \mathbb R\times\mathbb R$. Define a relation
$(a, b) ∼ (c, d) \iff b − a = d − c$.
Show that $∼$ is an equivalence relation and describe equivalence classes to $∼$ geometrically.
So, I write up the definitions of reflexive, symmetric and transitive properties and my intuition tells me that "yes, this seems ok". $(a,b)$ knows $(a,b)$, for all $a,b,c,d$ that is $(a,b) ∼(c,d)$ then $(c,d) ∼(a,b)$ and $(a,b) ∼(c,d) ∼ (e,f)$.
But is that enough, only listing the properties the relation has to follow to be equivalent, and by intuition determine that they do? Seems like I've solved nothing.
Edit: Thank you for the input people. Still struggling to get the "right" mindset for doing math this way, so I'm doubting pretty much everything I do.
Best Answer
Your intuition is correct, but is not a formal proof.
In this simple case, also a formal proof is simple. For the transitivity, as an example, you can write something as:
If $ (a,b) \sim (c,d)$ and $(c,d)\sim (e,f)$ than, by definition of $\sim$ we have: $$ b-a=d-c \qquad \land \qquad d-c=f-e $$ so, by the transitivity of $=$ in $\mathbb{R}$ we have: $$ b-a=f-e $$ and this means that $(a,b)\sim (e,f)$.