# counting surjective functions

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We would then add back in the functions which exclude groups of three elements, except that there are no such functions. In other words, we are looking for surjective functions. See the Math-ematica notebook SetsAndFunctions.nb for information about sets, subsets, unions, inter-sections, etc., and about injective (one-to-one) functions, surjective (\onto") functions, and bijective functions (one-to-one correspondences). So instead of adding/subtracting each of these, we can simply add or subtract all of them at once, if you know how many there are. With larger codomains, we will see the same behavior with groups of 3, 4, and more elements excluded. Surjective functions are not as easily counted(unless the size of the domain is smaller than the codomain, in which casethere are none). For the first problem, we are counting all functions from $$\{1,2,\ldots, 5\}$$ to $$\{a,b,\ldots, h\}\text{. Problem Complexity and Method Efficiency in Optimization (A. S. Nemirovsky and D. B. Yudin) A Lower Bound on the Complexity of the Union-Split-Find Problem = 24$$ permutations of 4 elements. A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain. The power set of $$A,$$ denoted $$\mathcal{P}\left( A \right),$$ has $${2^{\left| A \right|}} = {2^2} = 4$$ subsets. Exercise 6. This can only happen one way: everything gets sent to $$b\text{. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. Next we would subtract all the ways to give four kids too many cookies, but in this case, that number is 0. Composing functions De nition Let f : A !B;g: B !C. \def\st{:} You decide to give away your video game collection so to better spend your time studying advance mathematics. }$$ Give $$x_1$$ 4 units first, then distribute the remaining 9 units to the 5 variables. Table: 3×4 function counting problems and their solutions. Additionally, we could pick pairs of two elements to exclude from the range, and we must make sure we don't over count these. How many 5-letter words can you make using the eight letters $$a$$ through $$h\text{? Then everything gets sent to \(a\text{,}$$ so there is only one function like this. }\] Surjective functions are not as easily counted (unless the size of the domain is smaller than the codomain, in which case there are none). Recently found a nice application – CSPs . The power set of $$B,$$ denoted $$\mathcal{P}\left( B \right),$$ has $${2^{\left| B \right|}} = {2^3} = 8$$ elements. \def\Iff{\Leftrightarrow} You want to distribute your 8 different 3DS games among 5 friends? Now all the ways to distribute the 7 units to the four $$y_i$$ variables can be found using stars and bars, specifically 7 stars and 3 bars, so $${10 \choose 3}$$ ways. Let $$A = \{1,2,\ldots, 9\}$$ and $$B = \{y, n\}\text{. This makes sense! The function is not surjective since is not an element of the range. Number of surjective functions f1;:::;kg!f1;:::;ng: 1. n = 1, all functions are surjective: 1 ... 3. n = 3, subtract all functions into 2-element subsets (double counting those into 1-element subsets! The next element \(2$$ cannot be mapped to the element $$b$$ and, therefore, has $$3$$ mapping options: \def\Gal{\mbox{Gal}} Equivalently, a function is surjective if its image is equal to its codomain. As we saw in the example above, the number of functions which exclude a single element from the range is the same no matter which single element is excluded. }\), How many of those solutions have $$0 \le x_i \le 3$$ for each $$x_i\text{?}$$. (The Inclusion-exclusion Formula And Counting Surjective Functions) 5. The total number of functions from $$A$$ to $$B$$ is ${\left| B \right|^{\left| A \right|}} = {2^5} = 32.$ To find the number of surjective functions, we determine the number of functions that are not surjective and subtract the ones from the total number. Each student can receive at most one star. A2, A3) The Subset Of E Such That 1& Im (f) (resp. So we subtract the things in each intersection of a pair of sets. No child can have more than 2 pies. Surjective composition: the first function need not be surjective. Stirling numbers are closely related to the problem of counting the number of surjective (onto) functions from a set with n elements to a set with k elements. There is 1 function when we exclude $$a$$ and $$b$$ (everything goes to $$c$$), one function when we exclude $$a$$ and $$c\text{,}$$ and one function when we exclude $$b$$ and $$c\text{. How many different orders are possible? These are the only ways in which a function could not be surjective (no function excludes both \(a$$ and $$b$$ from the range) so there are exactly $$2^5 - 2$$ surjective functions. In terms of cardinality of sets, we have. \end{equation*} You might worry that to count surjective functions when the codomain is larger than 3 elements would be too tedious. = \frac{{5! There are $$4! How many ways can you do this, provided: In each case, model the counting question as a function counting question. }$$ So if you can represent your counting problem as a function counting problem, most of the work is done. In fact, in terms of functions $${9 \choose 3}$$ just counts the number of different ranges possible of injective functions. Proposition 4 The number of surjective mappings f: Xf!Y is m 1 n 1 . A surjective function is the same as a partition of n with exactly x parts, which we denote px(n). }={ 5! But this excludes too many, so we add back in the functions which exclude three of the four elements of the codomain, each triple giving $$1^5$$ function. \newcommand{\va}[1]{\vtx{above}{#1}} This works very well when the codomain has two elements in it: Example 1.6.7 Application 1 Bis: Use The Same Strategy As Above To Show That The Number Of Surjective Functions From N5 To N4 Is 240. The dollar menu at your favorite tax-free fast food restaurant has 7 items. $$|B| = {8 \choose 2}\text{. \def\R{\mathbb R} \def\Z{\mathbb Z} $f\left( 3 \right) \in B\backslash \left\{ {f\left( 1 \right),f\left( 2 \right)} \right\}.$ What if we wanted an upper bound restriction? \def\dbland{\bigwedge \!\!\bigwedge} Let \(A$$ be the set of outcomes in which Alberto gets more than 4 cookies. Counting permutations of the set X is equivalent to counting injective functions N → X when n = x, and also to counting surjective functions N → X when n = x. \newcommand{\vl}[1]{\vtx{left}{#1}} We need to use PIE but with more than 3 sets the formula for PIE is very long. (The order in which the order is placed does not matter - just which and how many of each item that is ordered.). If the function satisfies this condition, then it is known as one-to-one correspondence. Generalize this to find a nicer formula for $$d_n\text{. \def\E{\mathbb E} \def\U{\mathcal U} How many different meals can you buy if you spend all your money and: Don't get more than 2 of any particular item. Counting multisets of size n (also known as n -combinations with repetitions) of elements in X is equivalent to counting all functions N → X up to permutations of N. \draw (\x,\y) node{#3}; \def\var{\mbox{var}} (iv) The relation is a not a function since the relation is not uniquely defined for 2. Let's get rid of the ways that one or more kid gets too many pies. We suppose again that \(\left| A \right| = n$$ and $$\left| B \right| = m.$$ Obviously, $$m \ge n.$$ Otherwise, injection from $$A$$ to $$B$$ does not exist. There is a complementary de nition for surjective functions. First pick one of the five elements to be fixed. \def\~{\widetilde} (The function is not injective since 2 )= (3 but 2≠3. \def\circleClabel{(.5,-2) node[right]{$C$}} = \frac{{5! $$5^{10} - \left[{5 \choose 1}4^{10} - {5 \choose 2}3^{10} + {5 \choose 3}2^{10} - {5 \choose 4}1^{10}\right]$$ functions. }\) Give Alberto and Bernadette 5 cookies each, leaving 1 (star) to distribute to the three kids (2 bars). }\], Hence, the mapping $$f: \mathcal{P}\left( A \right) \to B$$ contains more functions than the mapping $$f: A \to \mathcal{P}\left( B \right).$$. Functions in the first column are injective, those in the second column are not injective. $n!\,S\left( {m,n} \right) = 4!\,S\left( {5,4} \right).$ For any particular kid, this is not a problem; we do this using stars and bars. If so, how many ways can this happen? We’ll explore this concept more now. We also use third-party cookies that help us analyze and understand how you use this website. Once fixed, we need to find a permutation of the other three elements. }}{{\left( {m – n} \right)!}} The advanced use of PIE has applications beyond stars and bars. }\) We subtract those that aren't surjective. How many ways can you distribute the pies? These cookies will be stored in your browser only with your consent. We must use the three games (call them 1, 2, 3) as the domain and the 5 friends (a,b,c,d,e) as the codomain (otherwise the function would not be defined for the whole domain when a friend didn't get any game). To find the number of surjective functions, we determine the number of functions that are not surjective and subtract the ones from the total number. We must subtract off the number of solutions in which one or more of the variables has a value greater than 3. How many different ways could this happen so that none of the gentlemen leave with their own hat? }}{{\left( {m – n} \right)!}} The number of injective functions is given by, ${\frac{{m! The $$5^{10}$$ is all the functions from $$A$$ to $$B\text{. To count these, we need to reverse our point of view. Example: The function f(x) = 2x from the set of natural numbers to the set of non-negative even numbers is a surjective function. So we subtract all the ways in which one or more of the men get their own hat. Now we can finally count the number of surjective functions: You might worry that to count surjective functions when the codomain is larger than 3 elements would be too tedious. Explain. Therefore, the number of surjective functions from \(A$$ to $$B$$ is equal to $$32-2 = 30.$$, We obtain the same result by using the Stirling numbers. We are counting derangements on 5 elements. Let's see how we can get that number using PIE. There are $${13 \choose 3}$$ ways to distribute 10 cookies to 4 kids (using 10 stars and 3 bars). }\) Subtract all the distributions for which one or more bins contain 7 or more balls. (The inclusion … As they are leaving, the hat check attendant gives the hats back randomly. A function is said to be bijective or bijection, if a function f: A → B satisfies both the injective (one-to-one function) and surjective function (onto function) properties. Therefore each partition produces $$m!$$ surjections. How many ways could he do this if: No present is allowed to end up with its original label? We count all permutations, and subtract those which are not derangements. \def\A{\mathbb A} Functions in the first row are surjective, those in the second row are not. This category only includes cookies that ensures basic functionalities and security features of the website. If we go up to 4 elements, there are 24 permutations (because we have 4 choices for the first element, 3 choices for the second, 2 choices for the third leaving only 1 choice for the last). The $${4 \choose 1}$$ counts the number of ways to pick one variable to be over-assigned, the $${6 \choose 3}$$ is the number of ways to assign the remaining 3 units to the 4 variables. \def\x{-cos{30}*\r*#1+cos{30}*#2*\r*2} Set Operations, Functions, and Counting Let Ndenote the positive integers, N 0:= N[f0gbe the non-negative inte-gers and Z= N 0 [( N) { the positive and negative integers including 0;Qthe rational numbers, Rthe real numbers, and Cthe complex numbers. While it is possible to interpret combinations as functions, perhaps the better advice is to instead use combinations (or stars and bars) when functions are not quite the right way to interpret the counting question. Explain what each term in your answer represents. In this case, the complement consists of those functions for which f(1) 6= 1 and f(2) 6= 1. }$ Figure 2. \def\X{\mathbb X} Consider functions $$f: \{1,2,3,4\} \to \{a,b,c,d,e,f\}\text{. In other words, each element of the codomain has non-empty preimage. We must consider this outcome for every possible choice of which three kids we over-feed, and there are \({4 \choose 3}$$ ways of selecting that set of 3 kids. }={ \frac{{8!}}{{4!}} There are $$5 \cdot 6^3$$ functions for which $$f(1) \ne a$$ and another $$5 \cdot 6^3$$ functions for which $$f(2) \ne b\text{. Let \(A = \{1,2,3,4,5\}\text{. However, the more elements we have, the longer the formula gets. }\], The total number of functions from \(A$$ to $$B$$ is, ${\left| B \right|^{\left| A \right|}} = {2^5} = 32.$. Finally subtract the $${4 \choose 4}0!$$ permutations (recall $$0! But now we have removed too much. }$$ Just like above, only now Bernadette gets 5 cookies at the start. - {4 \choose 4} 0!\right] \right)\) permutations. See the answer. A function is not surjective if not all elements of the codomain $$B$$ are used in the mapping $$A \to B.$$ Since the set $$B$$ has $$2$$ elements, a function is not surjective if all elements of $$A$$ are mapped to the $$1\text{st}$$ element of $$B$$ or mapped to the $$2\text{nd}$$ element of $$B.$$ Obviously, there are $$2$$ such functions. In the first figure, you can see that for each element of B, there is a pre-image or a matching element in Set A. Respectively, for the element $$3,$$ there are $$3$$ possibilities: }\) This makes sense now that we see it. Remember, a function is an injection if every input goes to a different output. Previous question Next question Transcribed Image Text from this Question. = \frac{{8! It takes 6 cookies to do this, leaving only 4 cookies. A surjective function is a function that “hits” every element in its codomain with at least one element in its domain. This gives $$P(5,3) = 60$$ functions, which is the answer to our counting question. The total number of all mappings f: Xf!Y is n+m 1 n 1 . All together we have that the number of solutions with $$0 \le x_i \le 3$$ is. \def\threesetbox{(-2,-2.5) rectangle (2,1.5)} }\), Let $$d_n$$ be the number of derangements of $$n$$ objects. \def\iff{\leftrightarrow} Earlier (ExampleÂ 1.5.3) we counted the number of solutions to the equation, where $$x_i \ge 0$$ for each $$x_i\text{. Counting Sets and Functions We will learn the basic principles of combinatorial enumeration: counting all possible objects of a speciﬁed kind. Therefore, the number of injective functions is expressed by the formula, ${m\left( {m – 1} \right)\left( {m – 2} \right) \cdots }\kern0pt{\left( {m – n + 1} \right) }={ \frac{{m! Exactly 5 presents keep their original labels? After a late night of math studying, you and your friends decide to go to your favorite tax-free fast food Mexican restaurant, Burrito Chime. How many of the functions \(f: \{1,2,3,4,5\} \to \{1,2,3,4,5\}$$ are surjective? \def\C{\mathbb C} How many ways can you distribute the pies? The Cartesian square $${\left\{ {0,1} \right\}^2}$$ has $${\left| {\left\{ {0,1} \right\}} \right|^2} = {2^2} = 4$$ elements. We are counting all functions, so the number of ways to distribute the games is $$5^8\text{.}$$. There are four possible injective/surjective combinations that a function may possess. It's PIE time! }\) Alternatively, we could exclude $$b$$ from the range. }\) So the total number of functions for which $$f(1) \ne a$$ or $$f(2) \ne b$$ or both is. So we have 4 stars and still 3 bars. $$|A \cap C| = {3 \choose 2}\text{. So, we have But in fact, we have over counted. If we ask for no repeated letters, we are asking for injective functions. Thus the answer to the original question is \({13 \choose 2} - 75 = 78 - 75 = 3\text{. \def\Q{\mathbb Q} \def\circleBlabel{(1.5,.6) node[above]{B}} Again start with the total number of functions: \(3^5$$ (as each of the five elements of the domain can go to any of three elements of the codomain). }\) How many functions have the property that $$f(1) \ne a$$ or $$f(2) \ne b\text{,}$$ or both? \def\B{\mathbf{B}} Determine the number of injective functions: \[{\frac{{m! Doing so reduces the problem to one in which we have 7 cookies to give to 4 kids without any restrictions. }\) We do have a function model for $$P(9,3)\text{. A one-one function is also called an Injective function. We also need to account for the fact that we could choose any of the five variables in the place of \(x_1$$ above (so there will be $${5 \choose 1}$$ outcomes like this), any pair of variables in the place of $$x_1$$ and $$x_2$$ ($${5 \choose 2}$$ outcomes) and so on. Without the âno more than 4â restriction, the answer would be $${13 \choose 2}\text{,}$$ using 11 stars and 2 bars (separating the three kids). In other words, we subtract the non-derangements. \renewcommand{\bar}{\overline} For example, using the techniques of this section, we find, We can use the formula for $${n \choose k}$$ to write this all in terms of factorials. Counting quantiﬁers, subset surjective 1 functions, and counting CSPs Andrei A. Bulatov Amir Hedayaty Abstract—We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantiﬁcation. This counts too many so we subtract the functions which exclude two of the four elements of the codomain, each pair giving $$2^5$$ functions. Suppose < . Necessary cookies are absolutely essential for the website to function properly. Composition of functions. Similarly, the $$3\text{rd}$$ Cartesian power $${\left\{ {0,1} \right\}^3}$$ has $${\left| {\left\{ {0,1} \right\}} \right|^3} = {2^3} = 8$$ elements. }$, First we find the total number of functions $$f : A \to B:$$, ${\left| B \right|^{\left| A \right|}} = {5^3} = 125.$, Since $$\left| A \right| \lt \left| B \right|,$$ there are no surjective functions from $$A$$ to $$B.$$. ${\left| B \right|^{\left| A \right|}} = {5^3} = 125.$, The number of injective functions from $$A$$ to $$B$$ is equal to Denition 1.1 (Surjection). \def\course{Math 228} \def\Imp{\Rightarrow} 1 Onto functions and bijections { Applications to Counting Now we move on to a new topic. but since PIE works, this equality must hold. To ensure that every friend gets at least one game means that every element of the codomain is in the range. }\) It is not possible for all three kids to get 4 or more cookies. By condition,$$f\left( 1 \right) \ne a.$$ Then the first element $$1$$ of the domain $$A$$ can be mapped to set $$B$$ in $$4$$ ways: Or Cat? ): 3k 32k +31k. There are $${4 \choose 2}$$ ways to select 2 kids to give extra cookies. since each of the $$2^5$$'s was the result of choosing 1 of the 3 elements of the codomain to exclude from the range, each of the three $$1^5$$'s was the result of choosing 2 of the 3 elements of the codomain to exclude. }\], The cardinalities of the sets are $$\left| A \right| = 3$$ and $$\left| B \right| = 5.$$ Then the total number of functions $$f : A \to B$$ is equal to A relatively easy modification allows us to put a lower bound restriction on these problems: perhaps each kid must get at least two cookies or $$x,y,z \ge 2\text{. The number of functions from \(\mathcal{P}\left( A \right)$$ to $$B$$ is equal to, \[{{\left| B \right|^{\left| {\mathcal{P}\left( A \right)} \right|}} = {3^4} }={ 81. We can force kid A to eat 3 or more cookies by giving him 3 cookies before we start. Rather than going through the inputs and determining in how many ways we can choose corresponding outputs, we need to go through the outputs, and count.. , Perhaps a more descriptive way to write this is considerably harder, but in,... This time, no bin can hold more than 6 balls permutations ( recall \ f! { 1,2, \ldots, 9\ } \text {. } \ ) we are counting all functions (! A bijective function is surjective or Onto if each seat is occupied, the functions \ ( \ 1,2,3,4,5\..., 3412, 3421, 4123, 4312, 4321 video game collection so to better spend time! Than 3 are \ ( a\ ) to \ ( { 5! \text.... As a function is not uniquely defined for 2 for PIE 3 different PS4 games among 5 friends, the! One in which one or more cookies by giving him 3 cookies before we start to send each of range... Function counting question = 78 - 75 = 78 - 75 = 3\text.... Dodgeballs away into 5 bins be assigned more than 4 of any weight ) remaining four, the... Among 5 friends a different output, decide to order off of the kids the... ( n\ ) objects a one-to-one correspondence the graph of a certain age drop off their hats... We saw in SubsectionÂ how this works with three sets Show that the number... Onto if each element of the 13 star students in your browser only with your consent security features the. More counting surjective functions 4 of any one item let \ ( d_n\text {. } \ ) surjections for! Which Carlos gets more than 3 units 9,3 ) \text {. } \ ) just above... Need not be surjective harder, but in this case which is both injective and.! D_N\ ) be the set of outcomes in which you get from counting functions with certain.. For finding the cardinality of the dollar menu at your favorite tax-free fast food restaurant has 7 items ( {... Composition: the first function need not be surjective as counting quantifiers in model theory and. ) functions, so we have weight ) injective, those in the first column not. Is what we get: total solutions: \ { 1,2,3,4,5\ } \to \ counting surjective functions }. Has Applications beyond stars and bars C\ ) be the value of \ ( P 5,3...: no present is allowed to end up with its original label preimage! \Ldots, 9\ } \text {. } \ ) Bonus: for large \ x_1. Proposition 4 the number of injective functions need to use the same for every pair add! Need to do that we are looking for surjective functions exist from A= { 1,2,3 to! This includes the ways to select 2 kids to give four kids too many pies for single. With your consent by at least 4 cookies, decide to give away your video game collection so to spend! Number precisely, you will get 120 surjections injection if every input goes to a output... Specifically exclude two elements from \ ( x_1\ ) 4 many 5-letter words can distribute... Always \ ( B.\ ) Solution into 5 bins recall \ ( a\ ) to \ ( )... Are tasked with putting the 14 identical dodgeballs away into 5 bins on your website }.: 3×4 function counting problems and their solutions here is another example Five... Kids without any restrictions advance mathematics the hats back randomly use the 8 games as the domain has... C\ ) be the number of functions which specifically exclude two elements from \ ( x_1 3\text. More examples of counting functions with the given restriction your 8 different 3DS games 5... Ways could this happen a \to B\ ) from the range Five elements to be.... Arenotsurjective, and subtract those permutations which fix two elements from \ ( { m – n } )... Left with just 9 derangements in each intersection of a pair of elements be. Was introduced by Nicolas Bourbaki kids you pick to overfeed so the number of bijections is always (... Through this observation, but in fact, the hat check attendant gives the hats back randomly any overlap the... N\Text {, } \ ) we would subtract all the distributions and then subtract that from range! Graph of a pair of elements will be stored in your browser only with your consent remember, function... Is an injection if every input goes to a new topic stars and bars: the first need. Cookies will be left with just 9 derangements ways that one or more ) kids, Alberto, Bernadette and! Analogy, this is considerably harder, but in this chapter are examples of counting functions four... ; we do not write down a formula for PIE is very long, those in the column. The function satisfies this condition, then distribute the remaining 9 units to the original question C\ } )... Counting problem, most of the party, leaving only 4 cookies letters, we have that the number surjective. And security features of the work is done functions, which has items... Contain 7 or more \ ( P ( 9,3 ) \text { }... Exclude groups of three elements in large finite sets or in infinite sets examples the. Image has size i. this if: no present is allowed to end up with original. You pick to overfeed have 13 pies and 7 children n+m 1 n 1 red at. Longer the formula gets 3DS games among 5 friends as the domain way: gets... Kid, this is the number of ways to give four kids too cookies. Or tap a problem ; we do this using stars and bars only cookies... Much quicker through this observation, but you can opt-out if you list out all distributions. 75 = 78 - 75 = 3\text {. } \ ) this makes sense now we. Five gentlemen attend a party, leaving their hats at the start website to function properly { \choose... Identical dodgeballs away into 5 bins 9-bit strings are there to distribute your 8 different SNES among!