I have been watching the YouTube series 'Start Learning Mathematics' by The Bright Side of Mathematics.
I am currently on episode #3 of the set series and he's just introduced us to 'ordered pairs.'
In more detail, the episode is about the Cartesian product, and how it is the set of all ordered pairs of sets A & B (at least this is what I understood from it). He gives the proper definition:
$$A\times B:= \{(a,b)\mid a\in A\ \wedge\ b\in B\}$$
And then he defines the ordered pair through Kuratowski's method/proof.
He starts by saying For elements $x,y$ write:
$$(x,y) := \{\{a\},\{a,b\}\}$$
and
$$(x,y)=(\tilde{x},\tilde{y}) \Leftrightarrow \{x\}=\{\tilde{x}\}\ \wedge\ \{y\}=\{\tilde{y}\}$$
$$\Leftrightarrow x = \tilde{x}\ \wedge\ y = \tilde{y}$$
It has left me very confused. I mostly understand the last bit, it seems obvious that if you have points $(a,b) = (c,d)$ then $a = c$ and $b = d$, but I still don't understand why the first set has $\{a,b\}$ in it and not just $\{b\}$, and what the entire point of these are.
I would appreciate any help, thank you!
Best Answer
Here are some words that people really should use a lot more in mathematics than they do: there are two things that people mean when they talk about "defining" a mathematical object, which you might call specifications and constructions (as far as I know there is no widely adopted language for this distinction in mathematics). A specification is like a blueprint; you specify the properties you want that object to have. A construction is like actually building a house; you actually construct the object to demonstrate that it really exists.
The specification of "ordered pair" is: whatever ordered pairs are, they should take as input two objects $a, b$ and return as output a third object $(a, b)$, and this third object should have the property that $(a, b) = (c, d)$ iff $a = c$ and $b = d$. In other words an ordered pair should be what it sounds like: it should contain exactly the information of the two objects that make it up, in order.
Kuratowski's "definition" of an ordered pair should more precisely be called a construction; it constructs, in set theory, an object satisfying the above specification. As mentioned in the comments, $\{ \{ a \}, \{ b \} \}$ does not satisfy the above specification, because $\{ \{ a \}, \{ b \} \} = \{ \{ b \}, \{ a \} \}$; this instead is a possible construction of an unordered pair.
Other constructions of ordered pairs are possible; see Wikipedia.