Функції як відображення множин

Functions and Relations

Relations are just rules that give one or more output for any given input.

You might recall that this is similar to functions which give one output for any input. Functions are a type of relation, but they specifically only produce $1$ output.

Function Example

An example of a function is $$y=2x$$ If you put in the input $\color{#8925ae}{x=1}$, you receive the output: $$\begin{align} y&=2x\\ y&=2({\color{#8925ae}{1}})\\ y&=2 \end{align}$$

Relation Example

An example of a relation would be $$y^2=x$$ If you put in $\color{#8925ae}{x=4}$, you receive the outputs: $$\begin{align} y^2&=x\\ y^2&=\color{#8925ae}{4}\\ y&=\pm\sqrt{4}\\ y&=2\text{ or }-2 \end{align}$$ We see $y$ can either be $2$ or $-2$. Both will make $4$ when squared. So this has multiple outputs.

(Video) Introduction to Functions

by mathman1024

This video by mathman1024 explains the basics of functions.

$\large{\text{Summary}}$

A relation is a set or collection of ordered pairs. Here's an example: $$\{(2,3),(4,1),(0,1),(2,-2),(-1,5)\}$$

The individual ordered pairs are the objects from each set.

Sometimes when we put a bunch of ordered pairs together, we get a pattern that creates a graph.

This equation graphed above is: $y^2=x$. We can define $y=x^2$ as a relation. Its inputs and outputs are a collection of ordered pairs. The domain is all of its $x$ values, and the range is all of its $y$ values.

It is also important to explain how functions tie in to this. A function is a just a subset, or type, of a relation. A function only has one output for any input. In other words, for every $x$-value you put into a function, it returns only one $y$-value.

Relations on the other hand, don't care. Given any input, they can have one or more outputs. In other words, for every $x$-value you put into a function, it returns one or more $y$-values.

Looking at the graph, when $x=4$ there are two possible $y$-values: $y=2$ and $y=-2$. This tells us the graph is of a relation.

We can tell the graphed equation is not a function is because it fails the vertical line test. It does, however, pass the horizontal line test.

Relations

There are a few ways of defining mathematical relations, and some are more complicated than others. Here is a simplified definition: Relations are relationships between sets of values. They map inputs to outputs. For our purposes, think of relations as mappings between the domain and the codomain.

One of the most important things to remember is that relations can map a given input to multiple outputs. Because of this, relations are not quite functions, because functions map each input to exactly one output. In other words, the set of all functions is a subset of the set of all relations - functions are always relations, but relations are not always functions.

Relations can be represented in a variety of ways: graphs, set mappings, ordered pairs, tables, etc. For example, say we have a relation that maps $x$ to $y$ if $x<y$ (the "less than" relation). Here are a few ordered pairs that are part of the relation: $$(1, 2)\\ (2, 5)\\ (2, 6)\\ (100, 10000)$$

The first number is always less than the second number here! Also, notice that this relation maps can map inputs to multiple outputs, which distinguishes it from a function. For example, take a look at $(2, 5)$ and $(2, 6).$ These are examples of how this relation maps an input of $2$ outputs greater than $2.$ There are actually infinitely many of these outputs!