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How does $\{a, b\}$ "become" $b$ in $(a, b) = \{\{a\}, \{a, b\}\}$?

If I substitute $a$ and $b$ with, say, $1$ and $2$, how is it explained that from $(1, 2) = \{\{1\}, \{1, 2\}\}$

$$2 = \{1, 2\}$$

arandompersonontheinternet
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  • ${a,b}$ is a way to rigorously write what you think of as $b \in (a,b)$ - this is just a formality to make the pair "ordered". In other words - it does "become" $b$ when you map from the formula to your mental picture. It is ${a, b}$ on paper but $b$ in your head. – Арсений Кряжев Jul 20 '22 at 16:53
  • @АрсенийКряжевiswithUkraine: you have missed the point of the question, I think. – TonyK Jul 20 '22 at 16:54
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    There is no reason to conclude from $(1, 2) = {{1}, {1, 2}}$ that $2 = {1, 2}$. You pulled this equality out of nowhere, so it’s impossible to say why your reasoning is wrong. – Mark Saving Jul 20 '22 at 17:03

2 Answers2

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In abstract mathematics we want to define things rigorously, and with set theory as the foundation of mathematics, everything is at the fundamental level a set. With this in mind, ask yourself "how would we define the ordered pair $(a,b)$ in terms of a set?". Well, the set $\{a, b \}$ is clearly not a suitable definition, because $\{ a,b \} = \{b, a \}$, so this would not give us an ordered pair, merely a pair. The set needs to impose an order on the two elements, i.e. a way to distinguish what is the first element and what is the second. Well, one way to distinguish them would be to append to the set $\{a, b \}$ a singleton indicating which of the two is the first element. This is the motivation behind the definition $$(a,b) = \{ \{a \}, \{a, b \} \}.$$ This definition includes the two elements present in the ordered pair and it indicates an order since one element (the first) occurs both in the singleton and in the two-element set.

It is important to note that this does not mean that $b = \{ a,b \}$ (no set may be a member of itself by the Axiom of Foundation). Here you should think of $a$ and $b$ as to given objects and the set $\{ \{a \}, \{a, b \} \}$ as your definition of what the expression $(a,b)$ means. A priori, the string $(a,b)$ has no meaning. We give it meaning through the definition above (often referred to as Kuratowski's construction).

I hope this clearifies the matter for you. If not, don't hesitate to leave a comment about what is still unclear.

Thusle Gadelankz
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  • Perfect answer, very clear, very starter-friendly, very well-explained. Thank you very much. I still don't get the concept of exactly why the singleton ${a}$ makes $a$ be the first element in $(a, b)$. I mean, of course ${{a}, {a, b}} \neq {{b}, {a, b}}$ and thus, $(a, b) \neq (b, a)$. Following this logic, is ${{a}, {a, b}, {a, b, c}}$ equal to the ordered triplet $(a, b, c)$? Is it because of how many times the variables appears in the set? The number of times implies the order? I'm really inexperienced for this mathematics, sorry. – arandompersonontheinternet Jul 20 '22 at 17:18
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    Have a look at his question : https://math.stackexchange.com/q/308422/305862 and as well https://math.stackexchange.com/q/4366850/305862 – Jean Marie Jul 20 '22 at 17:47
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    @arandompersonontheinternet, I see where you're coming from! However you still seem a bit confused about what's happening, and the definition you propose for $(a, b, c)$ doesn't work. What is going on is that in "actual maths", we know about a concept of ordered pair, where $(a, b) = (c, d)$ iff $a = c$ and $b = d$. In set theory, we would like to have such a notion - so we say that $(a, b)$ is a shorthand for some expression that has this property. The exact expression we choose is arbitrary and does not have a deep meaning - and you should not try to interpret it semantically... – Izaak van Dongen Jul 21 '22 at 11:13
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    In particular we could instead choose to define $(a, b) = {{b}, {a, b}}$ and this definition would be completely fine. Or we could define $(a, b) = {{\emptyset, a}, {{\emptyset}, b}}$ as this also has the property we want. Your definition for $(a, b, c)$ doesn't work as you end up having $(a, b, b) = (a, b, a)$. This is quite subtle! A better definition would be $(a, b, c) = ((a, b), c)$ or $(a, b, c) = {(0, a), (1, b), (2, c)}$. – Izaak van Dongen Jul 21 '22 at 11:13
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    @IzaakvanDongen thank you for your comments! – arandompersonontheinternet Jul 21 '22 at 11:26
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$(a,b)$ is an ordered set, which becomes $\{\{a\},\{a,b\}\}$ in terms of unordered sets through the convention that the $n$'th element of the ordered set becomes an unordered set with cardinality $n$ containing the $n$'th and all previous is the. The $=$ in this case represents that convention of considering ordered sets special cases of unordered sets.

MeMyselfI
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