Part (a): Counting Functions
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Understanding Surjective Functions
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Today, we will talk about surjective functions. Remember, a function is surjective if every element in the codomain gets mapped from at least one element in the domain. Can anyone give me an example?
What if we have a function f: A -> B, where A has 3 elements and B has 2 elements? Could that be surjective?
That's correct! As long as each element from B has at least one pre-image from A, it works. For instance, if f(1) = b1, f(2) = b1, and f(3) = b2, then every element in B has a pre-image, making it surjective.
Does that mean it also has to be injective?
Good question! Not necessarily. A surjective function can have multiple elements mapping to the same element in the codomain. Which brings us to injective functions…
So injective means one-to-one?
Exactly! If no two elements in the domain are mapped to the same element in the codomain, it's injective.
Can you have a function that's both surjective and injective?
Yes! That's called a bijective function. But remember, for a bijection, the sizes of the domain and codomain must be equal.
To wrap up this session, remember: surjective = onto, injective = one-to-one, and bijective = both, with equal sizes of their sets.
Counting Functions
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Now, let’s move on to counting functions. If you have two sets, X with m elements and Y with n elements, how many possible functions can you form?
Wouldn't it be n^m? Like, each element in X can map to any element in Y?
Yes! Great insight! The total number of functions from X to Y is n raised to the power of m, or n^m.
And how do we count injective functions from X to Y?
For injective functions, the process is different. You must reduce the choices as you assign images. It starts with n choices for the first, then n-1 for the second, and so on, giving us: n * (n - 1) * ... * (n - m + 1).
And for bijective functions, it’ll be the same?
Not quite! For bijections, we need n = m, and the count becomes n!
So, bijections relate to permutations of elements?
Exactly! To summarize, the number of functions is n^m, injective is n! / (n - m)!, and bijective is n! if m = n.
Stirling Function and Surjective Functions
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Next, let's talk about the Stirling function, which counts the number of ways to partition a set. Can someone explain how this relates to surjective functions?
Is it because each partition can represent the pre-images for elements in the codomain?
Exactly! Each subset in the partition can correspond to a unique image in Y. For example, if we partition set X into n subsets, that alignment helps count the surjective functions.
So if we want the number of surjective functions, we can use Stirling numbers?
Yes! The total count of surjective functions from set X to set Y is given by the number of partitions multiplied by the number of permutations of those partitions.
Can you remind us of the formula?
Of course! It’s S(m, n) * n!, where S(m, n) is the Stirling number of the second kind representing partitions.
So, in summary: Stirling functions help us count partitions, and these partitions help count surjective functions.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section covers how to count the number of functions between two sets, identifies conditions for injective and surjective functions, and elaborates on the significance of the Stirling function in counting the number of ways to partition sets. It emphasizes relationships between the cardinalities of sets and characteristics of the functions defined on them.
Detailed
Counting Functions
This section covers the essential aspects of counting functions, particularly focusing on three main types: surjective, injective, and bijective.
- Surjective Functions: A function is surjective (onto) if every element in the codomain has at least one pre-image from the domain. The example provided illustrates the potential lack of injectivity in surjective functions over infinite sets, leading to the need for careful definitions.
- Injective Functions: A function is injective (one-to-one) if distinct elements in the domain map to distinct elements in the codomain. The discussion highlights that deviating from injectiveness in function construction can lead to a misunderstanding of count outcomes.
- Bijective Functions: A bijection combines both properties, establishing a one-to-one correspondence between domain and codomain. The section stresses that the necessity of bijections requires equal cardinality in the addressing sets.
- Counting Methodologies: Using combinations and permutations, the section demonstrates how to compute the number of functions based on the sizes of the sets involved. This includes illustrating how bijections relate directly to permutations of set elements.
- Stirling Function: The concept of the Stirling function, particularly of the second kind, is introduced. This function counts the ways to partition a set into a specified number of non-empty subsets. Additionally, it's shown that the creation of surjective functions aligns with partitioning the domain set appropriately.
Through this section, the foundations of counting functions are laid, illustrating their implications in set theory and their applications in combinatorial contexts.
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Total Number of Functions
Chapter 1 of 3
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Chapter Content
You are given two sets X and Y consisting of m and n number of elements. So, I am calling the elements of X = {x₁, x₂, …, xₘ} and the elements of Y = {y₁, y₂, …, yₙ}. We are supposed to find out the number of functions from X → Y. It turns out that the number of functions will be n^m, why so? Because when we want to build a function from the set X → Y, each element xᵢ from the set X has to be assigned an image that is the definition of a function. Now how many ways I can assign an image for the element xᵢ? Well, I can assign y₁ as the possible image for xᵢ, I can pick y₂ as the possible image for xᵢ and so on. So, there are n possibilities when it comes to assigning an image for an element xᵢ. And the image for xⱼ and the image for xₖ are independently picked.
Detailed Explanation
In this section, we explore how to count the number of functions that can be formed from one set to another. If we have two sets, X with m elements and Y with n elements, we can create various functions from X to Y. A function requires assigning each element in set X an output from set Y. For each individual element in X, there are n choices from Y. Since each choice is independent of others, the total number of functions can be calculated using the formula n^m, where m is the number of elements in X. For example, if X has 2 elements and Y has 3 elements, then the number of possible functions is 3^2 = 9.
Examples & Analogies
Think of a school where there are 2 students (set X) and 3 different grades they could receive (set Y). Each student can independently receive any grade out of the 3 available. So, Student 1 can get A, B, or C, and Student 2 can also get A, B, or C. The total combinations of grades (functions) they can receive is 3 x 3 = 9 different possible outcomes.
Counting Injective Functions
Chapter 2 of 3
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Chapter Content
Part b asked you to find out the number of injective functions from the set X to the set Y. Now we cannot say that the images for every element xᵢ is chosen independently. Because now we are counting or we are interested in the injective functions, and in injective functions for every distinct element xᵢ from the set X you have to assign a unique image. You cannot have both x₁ and x₂ getting mapped to the same element in the Y set. So, that is why when it comes to selecting the image for x₁, I have n possibilities. But once I have decided the image for x₁, I cannot assign that image to be a possible image for element x₂, that is why for x₂ I have n - 1 possible images. Likewise, for the mth element from the set X, I can only assign n - m + 1 possible images. So, the total number of injective functions will be n × (n - 1) × (n - 2) × ... × (n - m + 1).
Detailed Explanation
Here, we look at injective functions, which are special because each element from the domain (set X) must map to a unique element in the codomain (set Y). This means that no two different inputs can produce the same output. Initially, for the first element from X, we have n choices in Y. However, for the second element in X, we now only have n - 1 choices, since we can't repeat the first choice. Continuing this pattern, for the third element we would have n - 2 choices, and so forth. Therefore, the formula for counting injective functions turns out to be the product of decreasing choices from n, which results in n × (n - 1) × ... until we've assigned all m elements from X. For instance, if X has 3 elements and Y has 5, the number of injective functions would be 5 × 4 × 3 = 60.
Examples & Analogies
Imagine a scenario where you have 3 employees (set X) competing for 5 different awards (set Y). Since no two employees can win the same award, the first employee has 5 award options. When it comes to the second employee, they have only 4 remaining choices (one award is already given out), and the third employee has 3 awards left to choose from. The total number of ways you could assign unique awards to the employees would be 5 × 4 × 3 = 60.
Counting Bijective Functions
Chapter 3 of 3
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Chapter Content
Part c asks you to find out the number of bijective functions from X to Y. The first thing to observe here is that for a bijection from X to Y we need |X| = |Y|. It is very easy to verify that if their cardinalities are different, then we cannot have a one-to-one and onto mapping. If the cardinality of X and Y is the same, then any bijection can be viewed as a permutation of the elements. For n elements in both sets, this means there are n! (n factorial) possible ways to map all elements from X to Y uniquely.
Detailed Explanation
This chunk covers bijections, where a function is both injective and surjective, meaning every element in set Y is paired with exactly one element in set X. The essential condition for a bijection is that both sets must have the same number of elements. Thus, if both X and Y have n elements, any bijective function can be reinterpreted as a unique arrangement (or permutation) of elements from set X to set Y. The total number of arrangements of n elements is n!, which is the factorial of n, calculated as n × (n - 1) × ... × 2 × 1. For example, if both sets contain 4 elements, the number of bijections is 4! = 24.
Examples & Analogies
Consider a classroom with 4 students (set X) and 4 unique chairs (set Y). If each student must sit in a different chair, you can think of all the possible seating arrangements as bijective functions. The ways to arrange the students in chairs is 4! = 24; each student can occupy any chair without sharing, producing multiple distinct arrangements.
Key Concepts
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Surjective Function: A function where every element from the codomain has at least one pre-image.
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Injective Function: A one-to-one function where different elements map to different images.
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Bijective Function: A function that is both injective and surjective, requiring equal cardinalities.
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Stirling Function: A function that represents the number of ways to partition a set into a given number of non-empty subsets.
Examples & Applications
An example of a surjective function can be from the set {1, 2, 3} to the set {a, b} where 1 maps to a, 2 also maps to a, and 3 maps to b.
The Stirling function S(3, 2) equates to 3, representing the partitions of a 3-element set into 2 non-empty subsets, e.g., {{1,2},{3}}, {{1,3},{2}}, and {{2,3},{1}}.
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Rhymes
Surjective and injective, two forms of fun, one maps all the way, the other one’s one!
Stories
Imagine a party where everyone must dance with someone. The surjective function ensures that everyone finds a partner to dance with, while the injective function ensures that nobody dances with more than one partner.
Memory Tools
SforS - Surjective Function reaches out to all in the 'S'et, while I in 'I'njective keeps it exclusive!
Acronyms
BIS = Bijection = Injective + Surjective; both needed to meet!
Flash Cards
Glossary
- Surjective Function
A function where every element in the codomain has at least one pre-image in the domain.
- Injective Function
A function where each element in the domain maps to a unique element in the codomain.
- Bijective Function
A function that is both surjective and injective, establishing a one-to-one correspondence.
- Stirling Function
A combinatorial function that counts the number of ways to partition a set into non-empty subsets.
- Cardinality
The number of elements in a set.
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