1.1.1: Extensionality
A set is a collection of objects, considered as a single object. The objects making up the set are called elements or members of the set. If \(x\) is an element of a set \(a\) , we write \(x \in a\) ; if not, we write \(x \notin a\) . The set which has no element s is called the empty set and denoted “ \(\emptyset\) ”.
It does not matter how we specify the set, or how we order its element s, or indeed how many times we count its element s. All that matters are what its element s are. We codify this in the following principle.
If \(A\) and \(B\) are sets, then \(A = B\) iff every element of \(A\) is also an element of \(B\) , and vice versa.
Extensionality licenses some notation. In general, when we have some objects \(a_{1}\) , …, \(a_{n}\) , then \(\{a_{1}, \dots, a_{n}\}\) is the set whose element s are \(a_1, \ldots, a_n\) . We emphasise the word “ the ”, since extensionality tells us that there can be only one such set. Indeed, extensionality also licenses the following:
\[\{a, a, b\} = \{a, b\} = \{b,a\}.\nonumber\]
This delivers on the point that, when we consider sets, we don’t care about the order of their element s, or how many times they are specified.
Whenever you have a bunch of objects, you can collect them together in a set. The set of Richard’s siblings, for instance, is a set that contains one person, and we could write it as \(S=\{\textrm{Ruth}\}\) . The set of positive integers less than \(4\) is \(\{1, 2, 3\}\) , but it can also be written as \(\{3, 2, 1\}\) or even as \(\{1, 2, 1, 2, 3\}\) . These are all the same set, by extensionality. For every element of \(\{1, 2, 3\}\) is also an element of \(\{3, 2, 1\}\) (and of \(\{1, 2, 1, 2, 3\}\) ), and vice versa.
Frequently we’ll specify a set by some property that its element s share. We’ll use the following shorthand notation for that: \(\{x : \phi(x)\}\) , where the \(\phi(x)\) stands for the property that \(x\) has to have in order to be counted among the element s of the set.
In our example, we could have specified \(S\) also as
\[S = \{x : x \text{ is a sibling of Richard}\}.\nonumber\]
A number is called perfect iff it is equal to the sum of its proper divisors (i.e., numbers that evenly divide it but aren’t identical to the number). For instance, \(6\) is perfect because its proper divisors are \(1\) , \(2\) , and \(3\) , and \(6 = 1 + 2 + 3\) . In fact, \(6\) is the only positive integer less than \(10\) that is perfect. So, using extensionality, we can say:
\[\{6\} = \{x : x\text{ is perfect and }0 \leq x \leq 10\}\nonumber\]
We read the notation on the right as “the set of \(x\) ’s such that \(x\) is perfect and \(0 \leq x \leq 10\) ”. The identity here confirms that, when we consider sets, we don’t care about how they are specified. And, more generally, extensionality guarantees that there is always only one set of \(x\) ’s such that \(\phi(x)\) . So, extensionality justifies calling \(\{x : \phi(x)\}\) the set of \(x\) ’s such that \(\phi(x)\) .
Extensionality gives us a way for showing that sets are identical: to show that \(A = B\) , show that whenever \(x \in A\) then also \(x \in B\) , and whenever \(y \in B\) then also \(y \in A\) .
Prove that there is at most one empty set, i.e., show that if \(A\) and \(B\) are sets without element s, then \(A = B\) .