What Is Equivalence?

(I’m putting this little post out because I want to do something more impressive, and I’ll need this lurking in the background. If it seems unmotivated, then, please treat it as a lemma of an essay.)

Most of us have a fairly decent idea of “equals”; at least, we’re fairly sure we know what it means to say “x is equal to y” and can draw from that other conclusions, such as, the controversial “y is equal to x”. Usually we get the idea early in learning arithmetic, and get used to it in working with numbers, and maybe stretch the idea (within the bounds of mathematics) to include things like two angles being equal or two shapes being equal.

Equivalence is a kind of generalizing of this equality idea: we’ll take a bit of what’s interesting about the idea of two things being equal, and use it in a new context. In this new context two things that might not be equal are still similar in some way that’s of interest for whatever we’re working on right now.

To write that “x is equal to y” efficiently we call on the equals sign and just put “x = y”. “x is equivalent to y” also begs for a shorthand notation, at least if you’re doing a lot of work with the idea, and the easiest way to type that is probably just to use a tilde: “x ~ y”, though I admit I prefer using a double tilde, “x ≈ y”, which isn’t too hard to do in HTML but is more work.

For two things to be equivalent you need to say equivalent with respect to what property. Sugar, sand, and salt are pretty much the same if all you’re interested in is how heaps of small-grained particles move; they’re not at all equivalent if you’re baking; and they’re only sort-of equivalent if you’re trying to melt sidewalk ice. You also need to say what set of things you’re drawing from; it’s very hard to answer whether sugar is equivalent to birds if you thought the discussion was about real numbers. Usually in practice the relationship — called the equivalence relation — carries with it an explicit statement of what the set of things is, unless it’s just blisteringly obvious from context.

To say that something is an equivalence relation means that it has to obey three rules, ones that look make it look a lot like ordinary old equality. The first is called reflexivity: any thing in the set is equivalent to itself. Any number equals itself; any article of clothing has the same color as itself; any person has the same gender as herself. Sounds like an unavoidably true property? Consider, for real numbers, the relationship “is less than”; there’s no number that is less than itself. “Is less than” can’t be an equivalence relationship.

The second is called symmetry: if one thing is equivalent to another, then, that other thing is equivalent to the first. If the number we’ve given the name “height” is equal to the number we’ve given the name “length”, so to does the number we’ve given the name “length” equal the number we’ve given the name “height”, and similarly good results can be found with shirts and people’s genders. For numbers, “Is less than” is ruled out right away; but the initially promising “Is less than or equal to”, which satisfies reflexivity, can flop on symmetry: 4 is less than or equal to 12, certainly, but not the other way around.

And the last is called transitivity: if one thing is equivalent to a second, and a second thing is equivalent to a third, then, the first thing has to be equivalent to the third. Ordinary old numbers being equal to one another are still transitive, and those shirts having the same colors work out too, and the same with people sharing a gender. Interestingly, both “Is less than” and “Is less than or equal to” are transitive, but since those fail on reflexivity or on transitivity they’re not equivalence relationships anyway.

There are a lot of equivalences out there, such as two geometric shapes being congruent, or for that matter just being similar (having the same shape but different sizes), or whole numbers having the same remainder when divided by, say, two (which is a fussy way of saying numbers are odd or are even), or two objects having the same temperature, or the like.