Operators

We found this rather pessimistic-sounding definition for Operator:

There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Compare the two definitions:

• A function takes a value and returns a value.
• An operator takes a function and returns a function.

As an example, the operator “Differentiation with respect to x” takes the function and returns the function .

If there are two operators to the left of a function, the operator closer to the function operates first:

where

The noun operator is synonymous with the noun functional.

Appendix A

We found this statement: “It is common in mathematics to use operator as a syntactic term and operation as a semantic term. Addition is an operation; the addition sign “+” is an operator symbol.”

Hmmm… what do “syntactic” and “semantic” mean–how do they contrast?

Appendix B

Sadly, the word Operator also gets used in a different way. The symbols for Addition and Multiplication are called binary operators.

A binary operator is an operation that combines two elements to produce another element.

Appendix C

A binary operation is also called a dyadic operation.