Operator

An op­er­a­tion $$f$$ on a set $$S$$ is a func­tion that takes some val­ues from $$S$$ and pro­duces a new value. An op­er­a­tion can take any num­ber of val­ues from $$S$$, in­clud­ing zero (in which case $$f$$ is sim­ply a con­stant) or in­finitely many (in which case we call $$f$$ an “in­fini­tary op­er­a­tion”). Com­mon op­er­a­tions take a finite non-zero num­ber of pa­ram­e­ters. Oper­a­tions of­ten pro­duce a value that is also in $$S$$ (in which case we say $$S$$ is closed un­der $$f$$), but that is not always the case.

For ex­am­ple, the func­tion $$+$$ is a bi­nary op­er­a­tion on $$\mathbb N$$, mean­ing it takes two val­ues from $$\mathbb N$$ and pro­duces an­other. Be­cause $$+$$ pro­duces a value that is also in $$\mathbb N$$, we say that $$\mathbb N$$ is closed un­der $$+$$.

The func­tion $$\operatorname{neg}$$ that maps $$x$$ to $$-x$$ is a unary op­er­a­tion on $$\mathbb Z$$: It takes one value from $$\mathbb Z$$ as in­put, and pro­duces an out­put in $$\mathbb Z$$ (namely, the nega­tion of the in­put). $$\operatorname{neg}$$ is also a unary op­er­a­tion on $$\mathbb N$$, but $$\mathbb N$$ is not closed un­der $$\operatorname{neg}$$ (be­cause $$\operatorname{neg}(3)=-3$$ is not in $$\mathbb N$$).

The num­ber of val­ues that the op­er­a­tor takes as in­put is called the ar­ity of the op­er­a­tor. For ex­am­ple, the func­tion $$\operatorname{zero}$$ which takes no in­puts and re­turns $$0$$ is a zero-ar­ity op­er­a­tor; and the op­er­a­tor $$f(a, b, c, d) = ac - bd$$ is a four-ar­ity op­er­a­tor (which can be used on any ring, if we in­ter­pret mul­ti­pli­ca­tion and sub­trac­tion as ring op­er­a­tions).

Parents:

• What’s the differ­ence be­tween an op­er­a­tor and an op­er­a­tion?

Also what’s the differ­ence be­tween an op­er­a­tion and a func­tion?