Tessellate S.T.E.M.S - Computer Science - School - Set 2 - Problem 1

In a certain programming language, the idea of sets are built-in. By sets, we refer to a kind of data structure that correspond to the mathematical notion of set, an unordered collection of distinct objects. In this programming language, a set containing the elements $x$ and $y$ is denoted $\{x, y\}$ ,

Certain developers need to implement the notion of an ordered pair in this programming language. An ordered pair is supposed to be a data structure that contains exactly two elements, not necessarily distinct, such that the elements are ordered. An ordered pair containing the two elements $x$ and $y$ , in that order , is denoted $(x,y)$ . By order, we mean that $(x, y) = (u,v) \implies x = u \text{ and } y = v$ .

Because the programming language only supports sets, the developers must implement ordered pairs in terms of sets. Here are a few suggestions they had:

1. $(a, b) := \{a, b\}$
2. $(a, b) := \{a, \{b\}\}$
3. $(a, b) := \{\,\{\{a\},\{\}\},\,\{\{b\}\}\,\}$
4. $(a, b) := \{\{a\}, \{a, b\}\}$

Which of these are valid, in the sense described above?

---

This problem is a part of Tessellate S.T.E.M.S.