Definition of sets This is a question based on what I have been learning about multi-variable calculus today. Basically I ask this because I'm not sure if mathematics learners use this term commonly.
Consider a multivariate function $f(x,y) = x^2+y^3-2y+4$, defined over the set $S=\left\{(x,y):x^2+y^2\leq1\right\}$, for example.
So we usually refer to some point that satisfies the set as $\begin{pmatrix}\frac 12, \frac 34\end{pmatrix}$. As I have more of an Information Systems background, I wonder if mathematics specialists refer to them as tuples? So in this case, I can say something like "If you consider all tuples, you get all points inside a unit circle if you observe from the $x$-$y$ plane.
This is something that has been grappling me...not sure if there are fellow computer science people that face this problem. This, along with others, are stumbling blocks because we have a way of thinking due to our coursework for years.
 A: In this case, we use the term ordered pairs to denote a 2-tuple, each point $(x, y)$ in the Cartesian plane. And yes, one could say the given set represents "All ordered pairs, or points, on or inside the unit circle in the $x$-$y$ plane.
For such a situation where the domain is a subset in Euclidean space, $\mathbb R^3$, we'd have ordered triplets (i.e., ordered 3-tuples) $(x, y, z)$ to denote each point in $R^3$ on which the domain is defined. 
For $R^n$, we define points as ordered n-tuples.
So there is no problem in thinking of ordered pairs,..., ordered $n$-tuples, but the qualifier ordered is crucial.  After all, $(1, 0) \neq (0, 1) \in \mathbb R^2$. Usually, however, ordered is presumed (or understood from context) and omitted.
A: In this particular instance, we would use "tuple", "pair", and "point" interchangeably. The usage may depend on some aspect of the problem the writer wishes to emphasize. e.g. is it important that it's a point in the plane? or is it important that we can split it into its two components?
Normally, we wouldn't say "(1/2,3/4) satisfies S", but "(1/2,3/4) is in S" or "(1/2,3/4) satisfies the condition $x^2+y^2\leq 1$". Also, there is an implied ambient space; in context, we would normally "all tuples" would refer to all points in the plane, and "all tuples in S" would refer to the (closed) unit disc.
