We know that a ring consists of a set equipped with two binary operations. My question is whether a ring is a set or not. For example, we can have $(\mathbb{R},+,-)$ where $\mathbb{R}$ is a set and $+$ and $-$ are binary operations associated with the set. Note that binary operations are functions, and functions are set, so we have a 3-tuple consisting of three sets. My first question is whether this tuple itself is a set? i.e. what exactly is a tuple?

In addition, the problem is I am not comfortable with defining ring as something with soemthing else. What exactly does it mean by "with"? (for example, is it a union?) it just seems overly informal.

Any help is apprecaited.