2.2: Defining sets

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There are two ways to define a set: extensionally and intensionally¹. I’m not saying there are two kinds of sets: rather, there are simply two ways to specify a set.

To define a set extensionally is to list its actual members. That’s what we did when we said $P = \{~\text{Dad}, \text{Mom}~\}$, above. In this case, we’re not giving any “meaning" to the set; we’re just mechanically spelling out what’s in it. The elements Dad and Mom are called the extension of the set $P$.

The other way to specify a set is intensionally, which means to describe its meaning. Another way to think of this is specifying a rule by which it can be determined whether or not a given element is in the set. If I say “Let $P$ be the set of all parents," I am defining $P$ intensionally. I haven’t explicitly said which specific elements of the set are in $P$. I’ve just given the meaning of the set, from which you can figure out the extension. We call “parent-ness" the intension of $P$.

Note that two sets with different intensions might nevertheless have the same extension. Suppose $O$ is “the set of all people over 25 years old" and $R$ is “the set of all people who wear wedding rings." If our $\Omega$ is the Davies family, then $O$ and $R$ have the same extension (namely, Mom and Dad). They have different intensions, though: conceptually speaking, they’re describing different things. One could imagine a world in which older people don’t all wear wedding rings, or one in which some younger people do. Within the domain of discourse of the Davies family, however, the extensions happen to coincide.

Fact: we say two sets are equal if they have the same extension. This might seem unfair to intensionality, but that’s the way it is. So it is totally legit to write: \[O = R\] since by the definition of set equality, they are in fact equal. I thought this was weird at first, but it’s really no weirder than saying “the number of years the Civil War lasted = Brett Favre’s jersey number when he played for the Packers." The things on the left and right side of that equals sign refer conceptually to two very different things, but that doesn’t stop them from both having the value 4, and thus being equal.

By the way, we sometimes use the curly brace notation in combination with a colon to define a set intensionally. Consider this: \[M = \{~k : \text{$k$ is between 1 and 20, and a multiple of 3}~\}.\] When you reach a colon, pronounce it as “such that." So this says “$M$ is the set of all numbers $k$ such that $k$ is between 1 and 20, and a multiple of 3." (There’s nothing special about $k$, here; I could have picked any letter.) This is an intensional definition, since we haven’t listed the specific numbers in the set, but rather given a rule for finding them. Another way to specify this set would be to write \[M = \{~3,6,9,12,15,18~\}\] which is an extensional definition of the same set.

Interesting thought experiment: what happens if you enlarge the intension of a set by adding conditions to it? Answer: increasing the intension decreases the extension. For example, suppose $M$ is initially defined as the set of all males (in the Davies family). Now suppose I decide to add to that intension by making it the set of all adult males. By adding to the intension, I have now reduced the extension from { Dad, T.J., Johnny } to just { Dad }. The reverse is true as well: trimming down the intension by removing conditions effectively increases the extension of the set. Changing “all male persons" to just “all persons" includes Mom and Lizzy in the mix.

Spelling nit: “intensionally” has an ‘s’ in it. “Intentionally,” meaning “deliberately,” is a completely different word.