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Chapter 3: Problem 44

Prove each. Any subset of a countable set is countable.

Short Answer

Expert verified
To prove that any subset of a countable set is countable, let A be a countable set and B 鈯� A. Define C as the set of natural numbers corresponding to the injection f: A 鈫� N. Define a function g: B 鈫� C such that g(x) = f(x) for all x in B. Since f is an injection and g(x) = f(x), g is also an injection. Thus, any subset of a countable set is countable, as it has an injection with the set of natural numbers.

Step by step solution

01

Definition of countable set

A set A is called countable if there exists an injective function f: A 鈫� N, where N is the set of natural numbers. This means that every element of A is uniquely mapped to a natural number. The set A can be finite or countably infinite (i.e. in one-to-one correspondence with the set of natural numbers).
02

Define the subset B and the set C

Let A be a countable set, and let B be a subset of A (B 鈯� A). Define C as the set of natural numbers such that C = f(B), where f is an injection from A to N. That is, C is the set of images of the elements of B under the function f.
03

Define a function g from B to C

Define a function g: B 鈫� C such that g(x) = f(x) for all x in B. Since f is an injection from A to N and B 鈯� A, this function g will map the elements of B to the corresponding elements of C.
04

Show that g is an injection

Now we need to show that g is an injective function, i.e. if g(x) = g(y) for x, y in B, then x = y. Recall that g(x) is defined as f(x) for all x in B. Therefore, if g(x) = g(y), then f(x) = f(y). Since f is an injection, x must equal y. Thus, g is an injective function from B to C.
05

Finalize the proof

In summary, we have defined a function g that maps B, a subset of a countable set A, to C, a set of natural numbers. We have also shown that g is an injective function from B to C. Therefore, any subset of a countable set is countable, as it has an injection with the set of natural numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Injective Functions
An injective function, also known as a one-to-one function, is a type of mapping between two sets where each element of the first set (domain) maps to a unique element of the second set (codomain). This means that no two different elements in the domain are mapped to the same element in the codomain.
To better understand injective functions, consider the following characteristics:
  • If for any elements \(x\) and \(y\) in the domain \(A\), it holds that \(f(x) = f(y)\), then it must be that \(x = y\).
  • Injective functions are important when discussing countable sets, because they ensure that every element is uniquely represented.
In the context of sets and countability, if you can map a set \(A\) injectively onto the natural numbers \(N\), then you are ensuring that each element in \(A\) is uniquely paired with a distinct natural number.
The Peculiarities of Natural Numbers
The set of natural numbers is often denoted by \(N\) and includes all positive whole numbers beginning from 1 (though sometimes starting from 0, depending on convention). These numbers are fundamental in mathematics because they form the basis for counting and ordering.
Key attributes of natural numbers include:
  • The property of being infinite which means there is no largest natural number; you can always add 1 to get the next natural number.
  • They provide a fundamental scale that many mathematical sets, when proven countable, can be mapped onto through injective functions.
In proofs involving countability, showing that a set \(A\) can be injectively mapped to \(N\) helps us understand its size in relation to the natural sequence of numbers, thus confirming its countability.
Finite and Infinite Sets Demystified
Sets in mathematics can be classified as either finite or infinite. It is crucial to understand the difference to grasp the concept of countability.
  • Finite sets are those with a definitive number of elements. For example, the set \(\{1, 2, 3\}\) has exactly three elements.
  • Infinite sets lack a clear endpoint. An example is the set of all natural numbers, which continues indefinitely without stopping.
When discussing countable sets:
  • A set is countable if there exists an injective function from it to the natural numbers, meaning you can list its elements similarly to how you list natural numbers.
  • Interestingly, a countable set can be either finite or infinite. For example, the set of even numbers is infinite yet still countable because you can pair each even number with a natural number.
Understanding these types and concepts of sets is fundamental in advanced mathematical studies, including proofs involving countability, illustrating how vast or constrained a set can be.

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