I need help with a problem from my discrete math course. I'm sure this problem is rather simple but I just can't figure out how to start it. I know I need to prove it is reflexive, symmetric, and transitive but I don't have any similar examples.
Let $\operatorname R$ be a relation on $\mathbb Z \times\mathbb Z$ such that $(a,b)\operatorname R(c,d) \iff a+b^3=c+d^3$.
Prove that $\operatorname R$ is an equivalence relation.
On
To prove R is an equivalence relation by definition is to prove it is reflexive, symmetric, and transitive
-Reflexivity: $(a,b)R(a,b) \iff a+b^3=a+b^3$ this one is trivial.
-Symmetry: $(a,b)R(c,d) \implies (c,d)R(a,b)$ this one is also trivial since it's the same equality.
-Transitivity: $(a,b)R(c,d) \land (c,d)R(e,f) \implies (a,b)R(e,f)$
$$\begin{align}(a,b)R(c,d)\land(c,d)R(e,f) &\iff (a+b^3=c+d^3) \land (c+d^3=e+f^3) \\&\iff a+b^3=e+f^3 \implies (a,b)R(e,f)\end{align}$$
Reflexivity: $\forall (a,b)\in\Bbb Z\times\Bbb Z : (a,b)\operatorname R(a,b)$ which means:
$$\forall (a,b)\in\Bbb Z\times\Bbb Z : a+b^3=a+b^3$$
Q: Does $a+b^3=a+b^3$ for every possible pair? Are there any possible exceptions?
A: Yes and no (resp.). So relation $\operatorname R$ is reflexive.
Symmetry: provide your definition. Does this hold for all two pairs? Are there any exceptions?
Transitivity: repeat the drill.