Defining an ordered pair as a set

Question

In mathematics we define mathematical objects in terms of other mathematical objects. For example, some textbooks define $(a,b)$ as a set. Such as, $$(a,b):= \{\{a\},\{a,b\}\}$$

Now, the cardinality of $\{\{a\},\{a,b\}\} $ is $2$. Does this imply that the cardinality of $(a,b)$ is $2$? I find it hard to speak of the cardinality of an ordered pair.

What I am asking here is why or why not can we speak of the cardinality of an ordered pair?

My question also applies to other objects such as relations as sets or ordered pairs, complex numbers as ordered pairs of real numbers... etc.

Why can we speak of a property of some objects and can't speak of the same property regarding another object that is defined in the terms of the first?

In summary, $$\left| {\left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\}} \right|=2$$

$$|(a,b)|=2?$$

Thanks in advance and sorry if my question is ambiguous.

Answer 1

You find it hard to talk about the cardinality of an ordered pair, because you don't think of an ordered pair as a set. You think about it as an ordered pair.

Similarly you will have a hard time talking about the cardinality of a real number. What is the cardinality of $\pi$? Well, of course it depends on how you construct the real numbers, and what exactly is $\pi$. But if you accept that $\pi$ is a particular set, then you can talk about its cardinality.

And the same principle applies to ordered pairs, or anything else really. If you accept that a particular object is represented by a particular set, then you can ask what are the elements of this set, and what is its cardinality, and so on.

As for whether or not $|(a,b)|=2$, this is true if and only if $a\neq b$. If $a=b$ then $(a,b)=(a,a)=\{\{a\}\}$ and its cardinality is $1$.