Problem 2 - A Primer of Real Analysis

Question

Let \(A\) be the set of even integers. Show that \(|A|=\aleph_{0}\).

Step-by-Step Solution

Answer

The cardinality of the set of even integers is \( \aleph_0 \).

Step 1: Understanding the Problem

We need to find the cardinality (size) of the set of all even integers, denoted as \(A\). The set of all even integers is \( \{ \, ..., -4, -2, 0, 2, 4, \,... \} \). We want to show that this set has the same cardinality as the set of natural numbers, which is denoted as \(\aleph_0\).

Step 2: Defining a Bijective Function

To prove that two sets have the same cardinality, we need to find a bijective function (one-to-one and onto) between the two sets. Let's define a function \( f: \mathbb{N} \to A \) where \( f(n) = 2n \). This function will map each natural number \( n \) to an even integer.

Step 3: Showing the Function is Injective

A function is injective if different elements in the domain map to different elements in the codomain. Assume \( f(n) = f(m) \). Then \( 2n = 2m \) implies that \( n = m \). Therefore, the function \( f(n) = 2n \) is injective.

Step 4: Showing the Function is Surjective

A function is surjective if every element in the codomain is mapped to by some element in the domain. Consider any even integer \(a\), which can be written as \(2m\) for some integer \(m\). Because \(m\) itself can be a non-negative integer (natural number), every even integer in \(A\) is the image of some natural number under \(f\). Thus, \(f\) is surjective.

Step 5: Conclusion

Since the function \( f(n) = 2n \) is bijective, we have shown there is a one-to-one correspondence between \( \mathbb{N} \) (the natural numbers) and \( A \) (the even integers). Therefore, the cardinality of \( A \) is \( \aleph_0 \).

Key Concepts

Cardinality

Cardinality is a concept in mathematics that helps us understand the size of a set. When we talk about the size, or cardinality, of a set, we're considering how many elements it contains. In real analysis and set theory, sets can be infinite, meaning they do not have a finite number of elements. For example, the set of natural numbers \( \{1, 2, 3, ...\} \) is infinite. Though infinite sets pose a unique challenge, mathematicians have developed ways to compare their sizes.

To express the size of infinite sets, we use cardinal numbers like \( \aleph_{0} \) to denote the smallest infinity, the size of the set of all natural numbers. When we say two sets have the same cardinality, it means they have the same number of elements, even if both are infinite. For the set of even integers \( A \), it has the cardinality \( \aleph_{0} \) because there exists a bijective function between the even integers \( A \) and the natural numbers. This shows that their sizes are effectively the same, albeit infinite.

Set Theory

Set theory is the study of collections of items or elements. These collections are called sets. Set theory provides us with a foundation for many areas of mathematics by helping us understand how to group and differentiate between elements. Sets can include anything, but in our context, we're often concerned with numbers.

Here are some important aspects of set theory:

• Elements: These are the objects or numbers that belong to a set. For example, the even integers set \( A = \{...,-4, -2, 0, 2, 4,...\} \).
• Subset: A set \( B \) is a subset of set \( A \) if every element of \( B \) is also an element of \( A \).
• Union and Intersection: The union of sets combines elements of two or more sets, while their intersection includes only those elements found in every set involved.

Understanding these concepts is crucial when dealing with different kinds of sets, like infinite sets. By analyzing the properties of sets, like the set of even integers, we gain deeper insights into their structure and cardinality.

Bijective Function

A bijective function is a special type of function in mathematics that plays a key role in proving that two sets have the same cardinality. For a function \( f: A \to B \) to be bijective, it must satisfy two conditions:

• Injective (One-to-One): This means each element of set \( A \) maps to a unique element in set \( B \). If \( f(a_1) = f(a_2) \), then \( a_1 = a_2 \).
• Surjective (Onto): Every element in set \( B \) must be able to be mapped from some element of set \( A \). Each item in \( B \) is an output of \( f \) from some input in \( A \).

In our specific example, the function \( f(n) = 2n \) is bijective between the natural numbers \( \mathbb{N} \) and the set of even integers \( A \). For each natural number \( n \), there is a corresponding even integer \( 2n \), showing injectiveness. Additionally, every even number in \( A \) can be represented as \( 2n \) for some natural \( n \), confirming surjectiveness. Thus, this bijective mapping proves that \( A \) and \( \mathbb{N} \) have the same cardinality of \( \aleph_{0} \).