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Understanding Cardinality
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Welcome everyone! Today we're diving into the concept of cardinality in sets. Can anyone tell me what cardinality refers to?
Is it how many elements are in a set?
Exactly! Cardinality measures the 'size' of a set in terms of its elements. For finite sets, it's just the count of those elements. What about infinite sets? How do we compare them?
I think we can use mappings to show if they’re the same size?
Right! We often use injective mappings to compare infinite sets. This brings us to the first sets we’ll examine today. What are our sets?
Introducing the Sets
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We have set A, which contains all real numbers between 0 and 1, excluding 0 and 1. And we have set B, which includes the numbers between 0 and 1, but includes 1. Can anyone summarize how we’ll prove they have the same cardinality?
We’ll show there are injective mappings from A to B and from B to A.
Correct! Let’s first create an identity mapping from set A to B. Who can tell me why this will work?
Because every number in A has a unique representation in B.
Exactly! Now, let's visualize the identity mapping before we move on.
The Injective Mapping from A to B
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When we apply the identity function, any element x in A maps to itself in B. Can someone explain why distinct elements remain distinct?
If x and y are different in A, their images are also different because we’re just mapping them directly.
Perfect! This part of the proof shows that the mapping is injective. Now, who remembers the function we use to map set B back to set A?
It’s g(x) = x/2, right?
Exactly! Why does this mapping help illustrate that B has elements that can fit into A?
Because it ensures that all outputs are still within the limits of set A.
Great explanation! This injective mapping completes our proof.
Conclusion of the Proof
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So, what can we conclude about the two sets based on our mappings?
They have the same cardinality!
Exactly! Through the injective mappings we've established, we adhere to the Schroder-Bernstein theorem, showing these two sets indeed have the same size. Does anyone have questions about cardinality or the methods we used?
Introduction & Overview
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Quick Overview
Standard
In this section, we explore the concepts of cardinality through the example of two sets of real numbers. By applying the Schroder-Bernstein theorem, we establish that both sets are equivalent in size, using injective mappings to illustrate the relationship between them.
Detailed
Detailed Summary
This section delves into proving that two specific sets of real numbers possess the same cardinality. The first set, denoted as set A, includes all real numbers in the interval (0, 1) but excludes 0 and 1. The second set, set B, includes all real numbers in the interval (0, 1]. To demonstrate that these two sets are of equal cardinality, we employ the Schroder-Bernstein theorem. This theorem posits that if there are injective (one-to-one) mappings from set A to set B and from set B to set A, the two sets must have the same cardinality.
We start by defining an injective mapping from set A to set B using the identity mapping, which maps each number to itself, effectively demonstrating that distinct elements in set A remain distinct when transferred to set B. For the reverse mapping, we define an injective function from set B back to set A, specifically the function g(x) = x/2. This function successfully maps every element in a way that still adheres to the restrictions of set A, thereby proving that both sets not only contain elements but align perfectly in terms of cardinality. The conclusion drawn is significant within the realm of discrete mathematics, aiding in the understanding of infinite sets and their relationships.
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Understanding the Sets
Chapter 1 of 5
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Chapter Content
We have two sets: the first set is the set of all real numbers between (0,1) but excluding 0 and 1. The second set is the set of all real numbers between (0,1] where 1 is included and 0 is excluded.
Detailed Explanation
The first step in comparing the two sets involves clearly defining them. The first set, denoted as Set A, includes all real numbers between 0 and 1, but importantly, 0 and 1 themselves are not part of this set. The second set, denoted as Set B, includes all real numbers in the same range but does include 1, while still excluding 0. Therefore, Set A can be written as (0, 1) and Set B as (0, 1]. This understanding is crucial for showing that both sets can be mapped to one another.
Examples & Analogies
Think of Set A as a playground that starts at the gate (0) and ends at the wall (1) but you can't step on either the gate or the wall. Set B is like a different playground that allows you to step on the wall (1) while still not allowing you to step on the gate. If everyone can play in their respective playgrounds but we want to see if you can swap players between the two playgrounds without anyone being left out, it helps visualize their sizes or cardinalities.
Applying the Schroder-Bernstein Theorem
Chapter 2 of 5
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Chapter Content
To prove the sets have the same cardinality, we can apply the Schroder-Bernstein theorem. This theorem indicates that if we can show there are injective mappings (one-to-one functions) from Set A to Set B and from Set B to Set A, then the two sets have the same size.
Detailed Explanation
The Schroder-Bernstein theorem is a fundamental result in set theory. An injective mapping from Set A to Set B means that each element in Set A can be paired with a unique element in Set B without any overlaps, which preserves the individuality of each element. By establishing two separate mappings—one from Set A to Set B and another from Set B back to Set A—we can demonstrate that both sets indeed share the same cardinality, or size. This approach is systematic and relies on understanding functions in mathematics.
Examples & Analogies
Imagine you have a box of toys (Set A) and a box of candy (Set B). If you create a unique pairing of each toy with a candy, ensuring that none are left without a partner, and then do the same for candies pairing back with toys, you can conclude that both boxes contain the same 'amount' of items despite seeming different at first glance.
Defining the Injective Mappings
Chapter 3 of 5
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Chapter Content
We define the first injective mapping f as the identity mapping where f(x) = x for x in Set A. The second mapping g is defined as g(x) = x/2 for x in Set B.
Detailed Explanation
The first mapping, labeled f, is straightforward since it directly pairs each element in Set A (between 0 and 1) with itself in Set B. This is called the identity mapping; it ensures that distinct real numbers in Set A correspond to distinct real numbers in Set B. The second mapping g, which takes any number in Set B and divides it by 2, maps the real numbers in Set B to Set A. By observing these mappings, we are effectively demonstrating that we can find unique partners for every number in both directions.
Examples & Analogies
Think of the first mapping as a personalized shopping service where each shopper takes only what they want (no changes), and the second mapping is like a discount service that reduces the price of items by half. The discount still ensures everyone can find their selection in both stores.
Verifying the Injective Mappings
Chapter 4 of 5
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Chapter Content
We verify that mapping f is injective since distinct inputs produce distinct outputs. Similarly for mapping g, when x ≠ 1, g(x) falls in (0, 1) and keeps x mapped appropriately. Even when x = 1, it correctly produces an output of 0.5 which is still allowed.
Detailed Explanation
To confirm that f is injective, we consider that if we take two different elements from Set A, their images under f will always be different. It's similarly true for mapping g because when we apply it to two distinct values, they will also yield different results. This is critical in proving the mappings do not overlap, thus establishing that both sets manage to maintain distinct identities. The definition of injective allows us to assert that the size of input and output sets remains consistent through these mappings.
Examples & Analogies
Imagine a library catalog! Each unique book (from Set A) maps to a unique location in the library (Set B) where it's stored. If someone finds a new shelf location (mapping) for a book, it doesn’t change the fact that each book remains distinct, ensuring the library never misplaces its collection. Similarly, a unique mapping keeps track of items effectively.
Concluding Cardinality
Chapter 5 of 5
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Chapter Content
With both injective mappings verified, we conclude that Set A and Set B have the same cardinality as per the Schroder-Bernstein theorem.
Detailed Explanation
Having demonstrated two injective mappings, we can confidently utilize the Schroder-Bernstein theorem to conclude that the two sets are indeed of the same size. This theorem serves as a powerful tool in set theory to compare the sizes of infinite sets, and by proving the existence of both mappings, we show mathematically that our initial intuition about Set A and Set B being comparable in terms of cardinality is correct.
Examples & Analogies
It’s like confirming balance in a scale. By weighing items from both sides (the mappings), we can affirm that despite appearances, both sides hold equal weight in their respective measurements. This also reinforces our understanding that not all infinite sets need to be numerically equal in size yet can perfectly match in pairs.
Key Concepts
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Injective Mapping: A type of function where distinct inputs in one set correspond to distinct outputs in another.
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Cardinality of Sets: Refers to either finite or infinite size of sets based on the number of elements.
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Schroder-Bernstein Theorem: A principle used to establish the equivalence of the cardinalities of two sets.
Examples & Applications
Set A = (0, 1) represents all real numbers between 0 and 1 excluding 0 and 1.
Set B = [0, 1] includes all real numbers between 0 and 1 including 1.
Memory Aids
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Rhymes
If injective's the way, distinct mappings by play; sets equal in size, let the cardinality rise.
Stories
Imagine two expanding circles, each representing a set. They grow larger, but as they overlap perfectly without edge touching, they maintain the same size – this illustrates cardinality equivalence.
Memory Tools
CIS for Cardinality: Count unique elements, Injective must match, Size equals success!
Acronyms
SIE for the sets
Set Identity Equals for proving same sizes.
Flash Cards
Glossary
- Cardinality
The measure of the 'size' of a set in terms of the number of elements contained within.
- Injective Mapping
A function between two sets where each element of the first set is associated with a unique element of the second set.
- SchroderBernstein Theorem
A theorem stating that if there exist injective mappings between two sets in both directions, then those sets have the same cardinality.
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