Question

Show that the two given sets have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula (not as a table). \(\mathbb{N} \times \mathbb{N}\) and \(\\{(n, m) \in \mathbb{N} \times \mathbb{N}: n \leq m\\}\)

Short answer

The bijection that can be defined between the sets \(\mathbb{N} \times \mathbb{N}\) and \{(n, m) \in \mathbb{N} \times \mathbb{N}: n \leq m\} can be given by the function f(n,m) = (n, m+n) if n <= m and f(n,m) = (n + m, m) if n > m.

Step-by-step solution

Understanding the sets

The first set, \(\mathbb{N} \times \mathbb{N}\), refers to the set of all ordered pairs of natural numbers. The second set, \{(n, m) \in \mathbb{N} \times \mathbb{N}: n \leq m\}, includes all such pairs where the first number is less than or equal to the second.

Constructing a bijection

The challenge lies in mapping every element of the first set to a unique element of the second set, and vice versa. To achieve this, a bijection function can be considered as follows: If (n,m) are elements of the first set, then the function f(n,m) can be defined as follows: (n, m+n) if n <= m and (n + m, m) if n > m. This function ensures that the first element of the ordered pair in the second set is always less than or equal to the second.

Confirming the bijection

Now check if this function represents a bijection. An effective bijection must be both injective (one-to-one) and surjective (onto). By construction, this is the case, since every pair (n,m) in the first set maps to a unique pair in the second set, and vice versa. It therefore establishes a bijection between the two sets.

Key concepts

Bijection

When we talk about the cardinality of sets, we're considering the 'size' of sets in terms of the number of elements within them. To show that two sets have the same cardinality, one effective method is to establish a bijection between them. A bijection is a special kind of function: it connects each element of one set to a unique element of another set in a one-to-one and onto fashion.

Creating a bijection between two sets proves they have an equal number of elements or the same cardinality. If you can match every element from one set to an element in the other without any left over, you've demonstrated a one-to-one correspondence, which is the essence of a bijection. This is fundamental in mathematics for comparing the sizes of infinite sets, such as the set of natural numbers.

Ordered Pairs

An ordered pair is a pair of elements where the order in which the elements appear is significant. In mathematics, ordered pairs are often written as \( (a, b) \), where \('a'\) is the first element and \('b'\) is the second element. This concept is crucial when working with Cartesian products of sets, where the set of all ordered pairs from two sets is formed.

For instance, the Cartesian product \(\mathbb{N} \times \( \mathbb{N} \)\) yields the set of all ordered pairs of natural numbers, and the order of numbers within each pair is important. The order allows us to consider properties like 'first' and 'second', which are essential in operations such as mapping and comparing.

Natural Numbers

The set of natural numbers, denoted as \(\( \mathbb{N} \)\), is one of the most basic and essential concepts in mathematics. It includes all positive integers starting from 1 and continuing indefinitely: 1, 2, 3, 4, 5, and so on.

Natural numbers are used for counting and ordering. In set theory and other branches of mathematics, they play a foundational role. When we create ordered pairs of natural numbers or apply functions to natural numbers, we are often investigating the properties of these fundamental building blocks of mathematics.

Injective Function

An injective function, also known as a one-to-one function, is a function with the special property that each element of the domain maps to a unique element of the codomain. This means there is no overlap: no two different elements from the domain are mapped to the same element in the codomain.

In formal terms, a function \('f: A \to B'\) is injective if, whenever \('f(x) = f(y)'\), then \('x = y'\). This uniqueness is crucial when proving the equality of set cardinalities through the establishment of bijections, ensuring that the mapping preserves distinctness among elements.

Surjective Function

A surjective function, sometimes called an onto function, is a function in which every element of the codomain is mapped to by at least one element of the domain. This coverage ensures that no element in the codomain is left without a pre-image.

For a function \('f: A \to B'\) to be surjective, each element \('b'\) in \('B'\) must have at least one corresponding element \('a'\) in \('A'\) such that \('f(a) = b'\). This condition is essential in establishing bijections, as it ensures that the domain is fully paired with the codomain, and thereby confirms that the two sets involved have the same cardinality.