Question

Write the following as English sentences. Say whether they are true or false. $$ \forall X \subseteq \mathbb{N}, \exists n \in \mathbb{Z},|X|=n $$

Short answer

The mathematical sentence translates to: 'For all subsets X of the set of natural numbers, there exists an integer n such that the size of X is n.' This statement is true.

Step-by-step solution

Translate the symbols to English

First, translate each mathematical symbol into an English equivalent. \( \forall \) stands for 'for all', \( \subseteq \) stands for 'subset of', \( \mathbb{N} \) represents the set of natural numbers, \( \exists \) stands for 'there exists', \( n \in \mathbb{Z} \) says that n is an element of integers, and \( |X|=n \) indicates that the cardinality (or size) of set X is n.

Form the English sentence

Next, combine these translations to form an English sentence. Our mathematical sentence converts to: 'For all subsets X of the set of natural numbers, there exists an integer n such that the size of X is n.'

Evaluate the truth value

Finally, consider whether this sentence is true or false. Given that every subset of natural numbers can indeed be assigned a unique cardinality (size) which is an integer (positive if the subset is finite, zero if it is empty and negative are not taken into consideration as they are not cardinal numbers), then this sentence is true. In fact, the cardinality of a set is always a nonnegative integer or infinity.

Key concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects called sets. A set can be defined explicitly by listing its elements or by specifying a property that its members must satisfy.

For instance, the set of natural numbers, denoted by \( \mathbb{N} \), includes all the counting numbers starting from 1, 2, 3, and so on. Sets can have an infinite number of elements, like \( \mathbb{N} \) does, or can be finite. The concept of subsets, denoted by \( \subseteq \), is also crucial in set theory. A subset of a set contains some or all elements of the parent set. An important aspect to understand about sets is the concept of cardinality, which refers to the number of elements in a set, symbolized as \( |X| \).

In summary, set theory helps us understand the nature of collections of objects and the relationships between them, laying the groundwork for more advanced mathematical concepts.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth or falsity of a mathematical statement. Proofs are essential for confirming the validity of theorems and propositions within mathematics.

In the context of the given exercise, proving that the statement is true involves demonstrating that for every subset of natural numbers, an integer can indeed represent its cardinality.

Proofs involve various techniques such as direct proof, contradiction, induction, and others. The process usually starts by making assumptions and then uses logical reasoning and theorems to reach a conclusion. Understanding the nature of mathematical proof is indispensable for students since it allows them to justify their solutions and ensures that mathematics remains a consistent and reliable discipline.

Natural Numbers

The set of natural numbers is denoted by \( \mathbb{N} \) and typically includes all positive integers starting from 1, which are used for counting and ordering. In some mathematical contexts, 0 is also included in the set of natural numbers.

The given exercise incorporates natural numbers through the notation \( \mathbb{N} \). When we talk about subsets of natural numbers, we refer to any collection of natural numbers, possibly all of them, or possibly an empty set which does not contain any elements.

Understanding natural numbers, their properties, and how they fit into broader mathematical concepts such as set theory and number theory is crucial for building a solid foundation in mathematics. Notably, natural numbers are infinite but countable, which means we can list them one by one, an attribute characterized by their 'natural' sequence.