Arithmetic operations on functions
Given two functions \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) (from the same set \( X \) to the reals), we can define their:
- sum, \( f + g : X \to \mathbb{R} \)
- \( (f + g)(x) := f(x) + g(x) \)
- difference, \( f - g : X \to \mathbb{R} \)
- \( (f - g)(x) := f(x) - g(x) \)
- maximum, \( \max(f, g) : X \to \mathbb{R} \)
- \( \max(f, g)(x) := \max(f(x), g(x)) \)
- minimum, \( \min(f, g) : X \to \mathbb{R} \)
- \( \min(f, g)(x) := \min(f(x), g(x)) \)
- product, \( fg : X \to \mathbb{R} \) (or \( f \cdot g \) )
- \( (fg)(x) := f(x)g(x) \)
- quotient, assuming \( g(x) \ne 0 \text{ for all } x \in X\), \( f/g : X \to \mathbb{R} \)
- \( (f/g)(x) := f(x)/g(x) \)
- constant multiple, \( cf : X \to \mathbb{R} \)
- \( (cf)(x) := cf(x) \)
There is really nothing special going on here. Not even any composition. All of these could apply to functions from some set to the rationals also. In general, when two functions share a domain and range, for every operation present in the range it should be possible to define a new function where each element of the domain is mapped to the value that is the result of the operation applied to the separate function outputs. Limitations may need to be applied, like in the case of the quotient.
I considered not adding this card, but I supposed that it would be good to become familiar with the symbols and comfortable with how rudimantary they are.