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Magazine Chapter 2: Problem 16
Show that a subset of a countable set is also countable.
Short Answer
Expert verified
A subset of a countable set is also countable because it is either finite or there is an injective function mapping it to the natural numbers.
Step by step solution
01
- Define Countable Set
A set is countable if it is either finite or has the same cardinality as the set of natural numbers \(\textbf{N}\). This means there exists a bijection (one-to-one correspondence) between the set and \(\textbf{N}\).
02
- Consider the Subset
Let \(\textbf{A}\) be a countable set and \(\textbf{B}\) be a subset of \(\textbf{A}\). Our goal is to show that \(\textbf{B}\) is also countable.
03
- Case 1: \textbf{B} is Finite
If \(\textbf{B}\) is finite, it is trivially countable by definition. So, \(\textbf{B}\) meets the criteria for being countable.
04
- Case 2: \textbf{B} is Infinite
If \(\textbf{B}\) is infinite, we need to show there is a bijection between \(\textbf{B}\) and \(\textbf{N}\) or a subset of \(\textbf{N}\).
05
- Injective Mapping
Since \(\textbf{A}\) is countable, there is a bijection \(f: \textbf{A} \to \textbf{N}\). Define a function \(g\) from \(\textbf{B}\) to \(\textbf{N}\) such that \(g(x) = f(x)\) for all \(x \in \textbf{B}\).
06
- Show Injective Nature of \(g\)
The function \(g\) is injective because if \(g(x_1) = g(x_2)\), then \(f(x_1) = f(x_2)\). Since \(f\) is a bijection, this implies \(x_1 = x_2\). Hence, \(g\) is also injective.
07
- Mapping to Natural Numbers
Since \(g \) is an injective function from \(\textbf{B}\) to \(\textbf{N}\), \(\textbf{B}\) has at most the cardinality of \(\textbf{N}\). Therefore, \(\textbf{B}\) is either finite or countably infinite.
08
- Conclude Countability
Combining both cases, whether \(\textbf{B}\) is finite or infinite, \(\textbf{B}\) is always countable.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Bijective Function
Understanding a bijective function is crucial in proving that a subset of a countable set is also countable.
A bijective function is a function that is both injective (one-to-one) and surjective (onto).
This means every element in the domain is mapped to a unique element in the codomain, and every element in the codomain is the image of exactly one element from the domain.
For example, consider the function \(f : \textbf{A} \to \textbf{N}\) where \(\textbf{A}\) is our countable set. A bijection ensures that for every element \(a \in \textbf{A}\), there exists a unique natural number \(n \in \textbf{N}\).
A bijective function is a function that is both injective (one-to-one) and surjective (onto).
This means every element in the domain is mapped to a unique element in the codomain, and every element in the codomain is the image of exactly one element from the domain.
For example, consider the function \(f : \textbf{A} \to \textbf{N}\) where \(\textbf{A}\) is our countable set. A bijection ensures that for every element \(a \in \textbf{A}\), there exists a unique natural number \(n \in \textbf{N}\).
Cardinality
Cardinality refers to the number of elements in a set, which helps in comparing the sizes of sets.
For countable sets, cardinality is either finite or the same as the set of natural numbers, denoted by \(\aleph_0\).
When a set \(\textbf{A}\) has the same cardinality as \(\textbf{N}\), it means there is a bijective function between \(\textbf{A}\) and \(\textbf{N}\), implying \(\textbf{A}\) is countably infinite.
Cardinality helps mathematicians understand whether a set is countable (finite or countably infinite) or uncountable.
For countable sets, cardinality is either finite or the same as the set of natural numbers, denoted by \(\aleph_0\).
When a set \(\textbf{A}\) has the same cardinality as \(\textbf{N}\), it means there is a bijective function between \(\textbf{A}\) and \(\textbf{N}\), implying \(\textbf{A}\) is countably infinite.
Cardinality helps mathematicians understand whether a set is countable (finite or countably infinite) or uncountable.
Natural Numbers
The set of natural numbers \(\textbf{N}\) is fundamental to defining countable sets.
Natural numbers \(\textbf{N}\) include all positive integers starting from 1, 2, 3, and so on, and sometimes include 0 depending on the definition.
In the context of countable sets, a set is countable if there exists a way to match every element of the set with a unique natural number, ensuring no elements are left unmatched on either side.
Natural numbers \(\textbf{N}\) include all positive integers starting from 1, 2, 3, and so on, and sometimes include 0 depending on the definition.
In the context of countable sets, a set is countable if there exists a way to match every element of the set with a unique natural number, ensuring no elements are left unmatched on either side.
Finite Set
A finite set has a limited number of elements. When discussing countability, if a subset \(\textbf{B}\) of a countable set \(\textbf{A}\) is finite, it is automatically countable.
This is because the set \(\textbf{B}\) can be enumerated explicitly, and no bijective function is needed to demonstrate countability.
For instance, if \(\textbf{B} = \{1, 2, 3\}\), it has a finite number of elements (three), making it trivially countable.
This is because the set \(\textbf{B}\) can be enumerated explicitly, and no bijective function is needed to demonstrate countability.
For instance, if \(\textbf{B} = \{1, 2, 3\}\), it has a finite number of elements (three), making it trivially countable.
Infinite Set
An infinite set has an unending number of elements. For a subset \(\textbf{B}\) of a countable set \(\textbf{A}\), if \(\textbf{B}\) is infinite, there should be a bijection between \(\textbf{B}\) and \(\textbf{N}\), or a subset of \(\textbf{N}\), to show it is countable.
Consider this function \(g: \textbf{B} \to \textbf{N}\) where each \(x \in \textbf{B}\) is mapped to a unique natural number via a bijection \(f\).
Through this mapping, we demonstrate that \(\textbf{B}\) has a cardinality that is countably infinite, asserting its countability.
Consider this function \(g: \textbf{B} \to \textbf{N}\) where each \(x \in \textbf{B}\) is mapped to a unique natural number via a bijection \(f\).
Through this mapping, we demonstrate that \(\textbf{B}\) has a cardinality that is countably infinite, asserting its countability.
