Difference between revisions of "Functional equation"
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==Advanced Topics== | ==Advanced Topics== | ||
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+ | ===Functions and Relations=== | ||
+ | Given a set <math>\mathcal{X}</math> and <math>\mathcal{Y}</math>, the Cartesian Product of these sets (denoted <math>\mathcal{X}\times \mathcal{Y}</math>) gives all ordered pairs <math>(x,y)</math> with <math>x \in \mathcal{X}</math> and <math>y \in \mathcal{Y}</math>. Symbolically, | ||
+ | <cmath> | ||
+ | \mathcal{X}\times \mathcal{Y} = \{ (x,y) \ | \ x \in \mathcal{X} \ \text{and} \ y \in \mathcal{Y}\} | ||
+ | </cmath> | ||
+ | |||
+ | A relation <math>R</math> is a subset of <math>\mathcal{X}\times \mathcal{Y}</math>. A function is a special time of relation where for every <math>y \in \mathcal{Y}</math> in the ordered pair <math>(x,y)</math>, there exists a unique <math>x \in \mathcal{X}</math>. | ||
===Injectivity and Surjectivity=== | ===Injectivity and Surjectivity=== |
Latest revision as of 16:59, 29 August 2024
A functional equation, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example, the following are functional equations:
Contents
Introductory Topics
The Inverse of a Function
The inverse of a function is a function that "undoes" a function. For an example, consider the function: . The function has the property that . In this case, is called the (right) inverse function. (Similarly, a function so that is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function is denoted by .
Intermediate Topics
Cyclic Functions
A cyclic function is a function that has the property that:
A classic example of such a function is because . Cyclic functions can significantly help in solving functional identities. Consider this problem:
Find such that . Let and in this functional equation. This yields two new equations:
Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have:
So, clearly,
Problem Examples
Advanced Topics
Functions and Relations
Given a set and , the Cartesian Product of these sets (denoted ) gives all ordered pairs with and . Symbolically,
A relation is a subset of . A function is a special time of relation where for every in the ordered pair , there exists a unique .
Injectivity and Surjectivity
Consider a function be a function from the set to the set , i.e., is the domain of and is the codomain of .
The function is injective (or one-to-one) if for all in the domain , if and only if . Symbolically, \begin{equation} f(x) \ \text{is injective} \iff (\forall a,b \in \mathcal{X}, f(a)=f(b)\implies a=b). \end{equation}
The function is surjective (or onto) if for all in the codomain there exists a in the domain such that . Symbolically,
\begin{equation}
f(x) \ \text{is surjective} \iff \forall a \in \mathcal{Y},\exists b \in \mathcal{X}: f(b)=a.
\end{equation}
The function is bijective (or one-to-one and onto) if it is both injective and subjective. Symbolically,
\begin{equation}
f(x) \ \text{is bijective} \iff \forall a \in \mathcal{Y},\exists! b \in \mathcal{X}: f(b)=a.
\end{equation}
The function has an inverse function , where , if and only if it is a bijective function.