Question

Find the following cardinalities. $$ \left|\left\\{x \in \mathbb{N}: x^{2}<0\right\\}\right| $$

Short answer

The cardinality of the set \{x ∈ ℕ: x² < 0\} is 0 as the set is an empty set.

Step-by-step solution

Identify the Content of the Set

This set contains numbers x with the property that x is a natural number and x² is less than 0. Natural numbers, by definition, are the set of positive integers, which start from 1 and go to infinity.

Analyze the Containment Condition

Next, look at the condition that for each x in the set, x² < 0 must hold. However, it is impossible for the square of a natural number to be less than zero - the square of any real number always 0 or positive - so there is no natural number that would satisfy this condition.

Derive the Cardinality

Thus, the given set is an empty set, which means it contains no elements. The cardinality of a set is defined as the number of elements in the set. Since the set here contains no elements, the cardinality of this set is 0.

Key concepts

Natural Numbers

The concept of natural numbers is fundamental to understanding many mathematical ideas, including set cardinality. Natural numbers are the numbers we use for counting and ordering in everyday life, starting from 1 and increasing by one at each step: 1, 2, 3, and so on, extending to infinity. They are denoted by the symbol \( \mathbb{N} \).

One important property of natural numbers to remember is that they are all positive and do not include zero. This is key when exploring sets and their elements, like in the given exercise where the set was defined using natural numbers.

Cardinality of a Set

Set cardinality is a term to describe the number of elements contained within a set. It can be thought of as a measure of a set's size, without consideration for what the elements actually are. Cardinality is often denoted using vertical bars, \(|A|\), which signify the cardinality of the set A.

For finite sets, cardinality is simply the count of distinct elements. For infinite sets, like the set of natural numbers, we use concepts from infinity to understand their unbounded size. In practical terms for exercises, determining if a set is finite or infinite helps in calculating cardinality. As seen in the exercise, sometimes an analysis of the conditions can lead to the realization that a set is empty and hence, has a cardinality of zero.

Empty Set

An empty set, also known as a null set, is a set that contains no elements. It is universally represented by the symbol \(\emptyset\) or \{\}.

The empty set is unique because it is the only set that has no members and its cardinality is always zero. Furthermore, it's a subset of every set because there are no elements in an empty set to contradict any condition for being a subset. Understanding the concept of an empty set is crucial in solving problems like our example where the impossibility of a natural number squared being negative leads us to an empty set.

Properties of Real Numbers

When investigating sets and their cardinalities, an awareness of the properties of real numbers is invaluable. Real numbers encompass the full spectrum of what might be termed the 'number line', including positive numbers, negative numbers, zero, rational numbers (like fractions), and irrational numbers (like \(\sqrt{2}\)).

Several key properties of real numbers are immediately applicable: for example, any real number squared is nonnegative, which tells us that no real number squared will ever be less than zero. This property was critical in the exercise as it led us to the understanding that the set of natural numbers fitting the given condition is actually an empty set.