Ordered pairs

 We've already introduced the notion of a pair of sets $\{x,y\}$. This object is symmetric insofar that it doesn't give a precedence to either $x$ or $y$. One way to formally state this is to observe that

$$\{x,y\}=\{y,x\}.$$

This follows simply from Extensionality: both sets have same elements! On the other hand, it will be of no surprise to you that sometimes we need an object which pairs together $x$ and $y$, while keeping the information which one comes first. Let us denote this ordered pair by $⟨x,y⟩$. According to our guiding principles, this should be another set, just another object in $V$. This means that, even though we have a clear idea what the object $⟨x,y⟩$ is, we need to pick a concrete way to code it within the universe. I say "pick" because there is more than one way to do so and giving the precedence to one over another is basically arbitrary.

So, let us make our choice and define the ordered pair of sets $x$ and $y$ to be the set

$$⟨x,y⟩:=\{\{x\},\{x,y\}\}.$$

Of course, at first this is just some random definition and we need to check that it behaves the way we would like it to. More precisely, we would like to have that for all sets $x,y,z,w$, it is true that

$$⟨x,y⟩=⟨z,w⟩⟹x=z\text{ and }y=w.$$

Compare this with the situation with ordinary (unordered) pairs: if $x$ and $y$ are two distinct sets, it holds that

$$\{x,y\}=\{y,x\},\text{ but }x\ne y\text{ and }y\ne x.$$

I'm not going to write down here the proof of this desired behavior of ordered pairs. The proof is just a tedious application of Extensionality and it would start like the following. We have that $\{x\}\in ⟨x,y⟩$ and that $⟨x,y⟩=⟨z,w⟩$, so we must have that $\{x\}\in ⟨z,w⟩$. Since the only elements of $⟨z,w⟩$ are $\{z\}$ and $\{z,w\}$, we get that either $\{x\}=\{z\}$ or $\{x\}=\{z,w\}$. In the first case we get $x=z$, while in the second we have $x=z=w$. This provides us with the fact that $x=z$, but it still leaves us to struggle to get $y=w$. You can now struggle on your own if you wish to. 😄