# Why is the word “complement” used in set theory?

Maybe this should have been on the English Exchange, but why do we use the word "complement" in set theory? If I have:

$$(A \cup B)'$$

Why does "complement" mean everything but the union? Is it because it is "all the things" that the original operation is not, thus it "completes" it?

I looked at the dictionary and wasn't sure.

Edit: $(A \cup B)'$ was only an example so I could use the complement mark. It was picked at random and was only meant to ask the question of what the word meant. It was not specifically about a union. I could have probably picked anything that allowed me to use the "tick mark" to indicate complement. My MathJax is not good and cumbersome for me, so I only used the single example.

• $A'$ would allow you to use the tick mark to indicate a complement, and it takes a lot less MathJax to write $A'$ than to write $(A \cup B)'.$ So it's still puzzling why you went to all the trouble to write something like $(A \cup B)'.$ But I suppose that's a moot point, since you already got the answer you needed. – David K Aug 8 '17 at 21:13
• " (A∪B)′ was only an example so I could use the complement mark." A FAR simpler and more pertanent example would be $A'$. Complement has nothing to do with unions so that is very misleading. Complment is simply about sets. So just use a set. – fleablood Aug 9 '17 at 3:10
• "My MathJax is not good and cumbersome for me, so I only used the single example." So why did you use a very hard example, instead of a very simple example. And why did you use MathJax at all. You don't need ANY MathJax to type A'. – fleablood Aug 9 '17 at 3:18

The first non-mathy definition that Merriam-Webster gives for "complement" is

something that fills up, completes, or makes perfect

You may also note that the other definitions have a similar connotation. Things complement each other if, when put together, they make up something that is complete. For another mathematical example, consider complementary angles, which add up to a "perfect" right angle.

In set theory, a set and its complement form a universal set (i.e. if $X$ is the universe, then $A \cup A^c = X$). Hence the two sets complement each other, in the sense that together, they make a whole.

Addendum: the Oxford English Dictionary (this may require academic access) indicates that "complement" was first used in the mathematical sense in Billingsley's 1570 translation of Euclid's Elements. At that time, it had been in English usage for at least 150 years (the earliest reference in the OED goes back to 1419, as far as I can tell, meaning something to the effect of "finishing or completing something"). The word originally comes from Latin: "complēmentum that which fills up or completes."

• From Latin complemens, present participle of complere, where com- = "together" and plere ="to make full", so "that which together (with the given) makes full" – Hagen von Eitzen Aug 9 '17 at 11:06
• It is worth pointing out that a complement is not well defined without the extra knowledge of what "universe" it refers to. Set theoretical complements are actually fairly rare, since the actual complement of a set (i.e. $A^\prime=\{x; x\not\in A\}$) is never a set in ZF(C). – DRF Aug 9 '17 at 11:46

Given how you talk about this, I suspect that part of your confusion is that you seem to think that the complement operator applies to some other operator (in your specific example, a union, but whatever other operator would end up there). But, complement is an operator that is applied to sets, not to operators. So, I can just have $A'$ as the complement of set $A$. And in the case of $(A \cup B)'$, the set to which you apply the complement operator happens to be the union of two sets.

So the complement is not, as you write, 'all the things that the original operator is not' (my emphasis) but rather 'all the things that the original set is not' (better put: 'all the things that are not in the original set'). Likewise, in your case, the complement does not 'complete' the operation of union, but simply 'completes' a set ... that again just happens to be a union of two sets.

• Union was just an example. Probably a poor one, but that was the intent (to be an example, not a poor one.) – johnny Aug 8 '17 at 16:42
• @Johnny So why did you say "all the things that the original operation is not"? And why not just use $A'$ as an example? – Bram28 Aug 8 '17 at 16:43
• "operation" could mean anything when I wrote it. I had to put something there as an example, so I picked that. It took me a while to format even that with MathJax, and I thought it was sufficient to get my question across. – johnny Aug 8 '17 at 16:44
• @Johnny But why not put a set there? Why not just $A'$? You say you had to put something there ... I am still concerned that you think that the complement applies to whatever operator you would put there ... – Bram28 Aug 8 '17 at 16:46
• @johnny OK, that's all good then ... sorry to pry! :) – Bram28 Aug 8 '17 at 16:48

"X complements Y" in colloquial English means basically "X has what Y lacks". This is exactly what the complement is in set theory, except that the complement of $A$ also has none of what $A$ has.

That said, my suspicion is that this term actually originates in mathematical French and was borrowed directly from mathematical French into mathematical English.

The complement of set $A$ in set $X$, written $X\setminus A$, is "the other half" of $A$. The set $X\setminus A$ is what the set $A$ "needs" in order to "be whole" again. In this context, "being whole" means being all of $X$, so $A$ together with $X\setminus A$ are whole.

More is true. There may be many sets that would make $A$ "whole" again, but $X\setminus A$ is the smallest of them all.

I pronounce the symbol "$X\setminus A$" as "$X$ without $A$", or "$X$ not $A$", or "$X$ punctured at $A$", or "the $X$-complement of $A$", or even "not $A$" when $X$ is clear from context.

The idea of complement is also very important in trigonometry (where $\pi/2-\theta$ is the $\pi/2$-complement of angle $\theta$), probability (where $1-p$ is the complement of probability $p$), and logic (where $\neg a$ is the complement of $a$ when both are elements of a complemented distributive lattice).