Clarification over an ambiguity in regards to the definition of functions.

Question

I have recently watched a couple of lectures in order to revise some of the notions that I will have to tackle in the coming months, and as I did so I have stumbled over something which baffled me a tad.

Now, within these lectures the lecturer described a relation as being a function solely if it maps one element from the domain to one sole, unique element from the codomain (or range assuming it is a surjective function we are dealing with); I found this a tad confusing since, ever since I can remember we considered functions to map a value from the domain to one sole value from the codomain/range, and if no value from the domain is mapped multiple times to the same value within the codomain/range then it is injective, hence abiding to the lecture's definition of a function.

Have I been missing something, do people automatically discard non-bijective, or at least non-injective functions for whatever reason?

Answer 1

Let $X,Y$ be sets and $\Gamma\subset X\times Y$ a subset. We call such a subset a relation from $X$ into $Y$. We then make the following standard definitions:

• $\text{dom}(\Gamma)=\{x\in X\,:\,\text{there is a $y\in Y$ such that $(x,y)\in \Gamma$}\}$, called the domain of $\Gamma$.
• $\text{im}(\Gamma)=\{y\in Y\,:\,\text{there is an $x\in X$ such that $(x,y)\in \Gamma$}\}$, called the image of $\Gamma$ (some people also call it the range of $\Gamma$, but I’d avoid this term because it is also sometimes used to refer to $Y$).
• We say $\Gamma$ is a function if for all $x\in X$, and all $y_1,y_2\in Y$, we have the implication \begin{align} \bigg((x,y_1)\in\Gamma\quad\text{and}\quad(x,y_2)\in\Gamma\bigg)\implies y_1=y_2. \end{align}
• Suppose $\Gamma$ is a function. We say it is injective if for all $x_1,x_2\in X$ and all $y\in Y$, we have the implication \begin{align} \bigg((x_1,y)\in\Gamma\quad\text{and}\quad(x_2,y)\in\Gamma\bigg)\implies x_1=x_2. \end{align}

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Here are some equivalent ways of saying things.

• $\Gamma$ is a function if and only if for all $x\in X$, there is at-most one $y\in Y$ such that $(x,y)\in \Gamma$.
• $\Gamma$ is a function if and only if for all $x\in\text{dom}(\Gamma)$, there is a unique $y\in Y$ such that $(x,y)\in\Gamma$.
• Suppose $\Gamma$ is a function. Then, $\Gamma$ is injective if and only if for all $y\in Y$, there is at-most one $x\in X$ such that $(x,y)\in\Gamma$.
• Suppose $\Gamma$ is a function. Then, $\Gamma$ is injective if and only if for all $y\in\text{im}(\Gamma)$ there is a unique $x\in X$ (which necessarily belongs to $\text{dom}(\Gamma)$) such that $(x,y)\in\Gamma$.

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So, you see the uniqueness is referring to different things. In the definition of function, the uniqueness is referring to the value in the codomain, i.e what happens in $Y$. In the definition of injectivity for functions, the uniqueness is referring to the domain, i.e what happens in $X$.