<p>Consider how you have defined the problem.</p>
<p>There is a function defined as
<span class="math-container">$$f: \mathbb {R} \rightarrow \mathbb {R}$$</span></p>
<p><span class="math-container">$$\;\;\;\;\;\;\;\;\;\;\;\;y=f(I)=I\qquad (1)$$</span></p>
<p>Look at the <strong>domain</strong> and the <strong>codomain</strong>: they are <strong>both <span class="math-container">$\mathbb {R}$</span></strong>, that is, according to the definition of function,  <span class="math-container">$f$</span> assigns one (and only one) real number to each real number.</p>
<p>Then <span class="math-container">$f$</span>, according to <span class="math-container">$(1)$</span>, is defined as the function that assigns to each value in <span class="math-container">$\mathbb {R}$</span>, that is a real number,  of the independent variable <span class="math-container">$I$</span> that real number itself.</p>
<p>This kind of function, of a set <span class="math-container">$A$</span> to itself, that assigns to each element of the domain the element itself is called the <strong>identity</strong>, or the <strong>identity function</strong>.</p>
<p>In your case, <span class="math-container">$f$</span> is the identity function between real numbers, that is <strong>the equality</strong> <strong>is between real numbers</strong>.</p>
<p>In your  economic problem, where  domain and codomain are <span class="math-container">$\mathbb{R}$</span>,  these real numbers  are some kind of <strong>measure, expressed through real numbers</strong>,  of input and output.</p>
<p>So, the equality has nothing to do with the nature of the objects involved, be they  inputs, outputs, oranges or apples. The objects are not necessarily the same, even if their measures, as expressed through real numbers, can be the same.</p>
<p>You can see:</p>
<p><a href="https://en.wikipedia.org/wiki/Identity_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Identity_function</a></p>

CC BY-SA 4.0
2023-06-22T07:54:35.793Z
2023-06-26T01:01:06.487Z
2023-06-26T01:01:06.487Z
BakerStreet
<p>A bijection between two things does not mean that these two things are the same.</p>
<p>For each night there is exactly one day, yet night and day are quite objectively different (e.g., in average temperature).</p>
<p>I can always transform 1 push-up's worth of energy (approx. 7kcal, but kcal is also a made up unit of measurement) into a push-up; yet the energy and the push up are IMO obviously quite different, for example the first could be transformed into many things.</p>
<p>Suppose there is an economy with two industries, producing apples and oranges, the production functions of which are <span class="math-container">$F_A(L) = L$</span> and <span class="math-container">$F_O(L) = L$</span>; i.e. in both industries the amount of output is equal to the amount of labor used. Yet clearly, labor and output are different, as it is possible that the 10 units of labor available are distributed so that the outputs in the industries are 8-2 respectively; thus neither output is equal to total labor. It is possible to transform an apple into an orange by reallocating labor, but apples are not oranges.</p>
<hr />
<p>Bonus: Rational numbers and integer numbers are not the same, though a bijection exists. Admittedly this is more tricky when infinite quantities are involved.</p>

CC BY-SA 4.0
2022-09-22T20:24:18.457Z
2022-09-22T20:24:18.457Z
Giskard
<p>Consider for example a production function with one variable.<br />
Let’s say the function is f:R-&gt;R y=I, where y is the quantity produced and I is the quantity of some input used for production.<br />
In this situation the quantity of some good produced and the quantity of some input used for production is always the same.<br />
We still would say there are two quantities/variables but what’s the difference between them when they are always the same?</p>

CC BY-SA 4.0
2022-09-22T16:26:44.143Z
Why would quantities always be the same? 2 apples = 100ml of juice. 20 apples = 1L of juice?
BrsG
CC BY-SA 4.0
2022-09-22T20:16:59.587Z
@BrsG That is an assumption. Suppose a relationship exists where two variables are the same.
Giskard
CC BY-SA 4.0
2022-09-23T13:52:20.203Z
@Giskard: I see, quantities=variables. This threw me off.
BrsG
<definition>
Difference between „equal variables“
Nige
# Difference between „equal variables“

(Asked by: username_0 on 2022-09-22)

Consider for example a production function with one variable.  
Let’s say the function is f:R->R y=I, where y is the quantity produced and I is the quantity of some input used for production.  
In this situation the quantity of some good produced and the quantity of some input used for production is always the same.  
We still would say there are two quantities/variables but what’s the difference between them when they are always the same?

**Tags:** definition

> **Comments:**
> **username_1**: Why would quantities always be the same? 2 apples = 100ml of juice. 20 apples = 1L of juice?  
> **username_2**: @username_1 That is an assumption. Suppose a relationship exists where two variables are the same.  
> **username_1**: @username_2: I see, quantities=variables. This threw me off.  

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## 2 answers

### Answer by username_3 on 2023-06-22. Score: 1

Consider how you have defined the problem.

There is a function defined as
$$f: \mathbb {R} \rightarrow \mathbb {R}$$

$$\;\;\;\;\;\;\;\;\;\;\;\;y=f(I)=I\qquad (1)$$

Look at the **domain** and the **codomain**: they are **both $\mathbb {R}$**, that is, according to the definition of function, $f$ assigns one (and only one) real number to each real number.

Then $f$, according to $(1)$, is defined as the function that assigns to each value in $\mathbb {R}$, that is a real number, of the independent variable $I$ that real number itself.

This kind of function, of a set $A$ to itself, that assigns to each element of the domain the element itself is called the **identity**, or the **identity function**.

In your case, $f$ is the identity function between real numbers, that is **the equality** **is between real numbers**.

In your economic problem, where domain and codomain are $\mathbb{R}$, these real numbers are some kind of **measure, expressed through real numbers**, of input and output.

So, the equality has nothing to do with the nature of the objects involved, be they inputs, outputs, oranges or apples. The objects are not necessarily the same, even if their measures, as expressed through real numbers, can be the same.

You can see:

<https://en.wikipedia.org/wiki/Identity_function>


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### Answer by username_2 on 2022-09-22. Score: 0

A bijection between two things does not mean that these two things are the same.

For each night there is exactly one day, yet night and day are quite objectively different (e.g., in average temperature).

I can always transform 1 push-up's worth of energy (approx. 7kcal, but kcal is also a made up unit of measurement) into a push-up; yet the energy and the push up are IMO obviously quite different, for example the first could be transformed into many things.

Suppose there is an economy with two industries, producing apples and oranges, the production functions of which are $F\_A(L) = L$ and $F\_O(L) = L$; i.e. in both industries the amount of output is equal to the amount of labor used. Yet clearly, labor and output are different, as it is possible that the 10 units of labor available are distributed so that the outputs in the industries are 8-2 respectively; thus neither output is equal to total labor. It is possible to transform an apple into an orange by reallocating labor, but apples are not oranges.

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Bonus: Rational numbers and integer numbers are not the same, though a bijection exists. Admittedly this is more tricky when infinite quantities are involved.


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