Please check my proof; how many functions exist from X to Y?

• SithsNGiggles
In summary, the conversation discussed the number of functions that exist from a set X to a set Y, with the formula m^n being determined as the possible number of functions. It was also noted that for every element in X, there is a unique element in Y that it can be paired with, resulting in n coordinate pairs in the function.
SithsNGiggles

Homework Statement

I just need to know if it makes sense; I was told that I can't have anyone make any improvements on what I've written myself.

Question: If $X = \{ x_1 , \ldots, x_n \}$ and $Y = \{ y_1 , \ldots, y_m \}$, how many functions from $X$ to $Y$ exist?

My answer: $m^n$ functions

The Attempt at a Solution

For any element $x \in X$, there exists a unique $y \in Y$ for which $F(x) = y$.

Every $n$ element in [itex[X[/itex] will be paired with anyone of the $m$ elements in $Y$.
i.e. there exist $m$ possible $F(x_1)$ in $Y$ that can be paired with $x_1$.
$x_2$ can be paired with $m$ possible $F(x_2)$
$\vdots$
$x_n$ can be paired with $m$ possible $F(x_n)$.

Because the domain $D_F = X$, every function generated through F will contain $n$ coordinate pairs. Furthermore, since there are $m$ possible values $F(x) = y$ for each element $x$, there are $n$ factors of $m$, or $m^n$, possible functions.

Thanks for any commentary (but not actual help!) you can provide.

Makes perfect sense to me.

1. How do you determine the number of functions from X to Y?

The number of functions from X to Y can be determined by using the formula 𝟮^𝒙𝒚, where x is the cardinality (number of elements) of set X and y is the cardinality of set Y. This formula applies when the function is defined as a mapping from every element in X to every element in Y, including the possibility of repeated values.

2. Can there be an infinite number of functions from X to Y?

Yes, there can be an infinite number of functions from X to Y if either set X or set Y has an infinite cardinality. This means that there are an infinite number of possible mappings between the two sets.

3. What is the difference between a one-to-one function and an onto function?

A one-to-one function, also known as an injective function, is a function where each element in the domain (set X) is mapped to a unique element in the codomain (set Y). An onto function, also known as a surjective function, is a function where every element in the codomain (set Y) is mapped to by at least one element in the domain (set X).

4. How does the cardinality of set X impact the number of functions from X to Y?

The cardinality of set X directly impacts the number of functions from X to Y. The larger the cardinality of set X, the larger the possible number of functions from X to Y. This is because each element in X can be mapped to any element in Y, and the more elements in X, the more possible mappings there are.

5. Can the number of functions from X to Y be negative?

No, the number of functions from X to Y cannot be negative. Functions are defined as a relation between two sets where each input (element in X) is mapped to a unique output (element in Y), and the number of functions represents the number of possible mappings. Therefore, the number of functions can only be a non-negative integer.

Replies
4
Views
3K
Replies
5
Views
1K
Replies
0
Views
331
Replies
2
Views
2K
Replies
3
Views
425
Replies
6
Views
995
Replies
15
Views
2K
Replies
8
Views
733
Replies
3
Views
2K
Replies
4
Views
517