Question

Find the following cardinalities. $$ \left|\left\\{x \in \mathbb{N}: x^{2}<10\right\\}\right| $$

Short answer

The cardinality of the set is 3.

Step-by-step solution

Understand the set notation

The set is \(\{x \in \mathbb{N}: x^{2}<10\}\). This means the set includes all the natural numbers \(x\) for which \(x^{2}<10\). The notation \(x \in \mathbb{N}\) means \(x\) is a natural number.

Find the natural numbers satisfying the condition

In this step, find out the natural numbers which when squared would be less than 10. These numbers are 1, 2 and 3 since \(1*1=1\), \(2*2=4\), \(3*3=9\) and \(4*4=16\) which is not less than 10. So, the set of numbers which fulfill the condition are \{1, 2, 3\}.

Calculate the cardinality

Now, find the cardinality of the set. The set consists of numbers \{1, 2, 3\}. So, there are 3 elements in the set.

Key concepts

Understanding Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as elements or members of the set. In the context of the given exercise, the set is defined as \( \{x \in \mathbb{N}: x^{2}<10\} \) which means that we are looking at all the elements, specifically natural numbers, which squared are less than 10.

When we talk about the cardinality of a set, we refer to the number of elements within that set. In simpler terms, it's the count of distinct objects it contains. For the exercise's set, the task is to identify the elements that meet the given condition and then simply count them to find its cardinality. Cardinality is symbolized by using vertical bars around the set, like this: \( |S| \) for set S.

Natural Numbers and Their Properties

Natural numbers are the numbers we use for counting and ordering. They are the numbers 1, 2, 3, and so on, extending to infinity. In the set notation \( \{x \in \mathbb{N} \) the symbol \( \mathbb{N} \) denotes the set of natural numbers.

It's important to note that natural numbers do not include zero or negative numbers. In the solution process of the given exercise, we examined the natural numbers sequentially to determine which ones, when squared, would result in a number less than 10. This operation must be done within the realm of natural numbers — fractions, decimals, and irrationals are not considered since they fall outside the set \( \mathbb{N} \).

Inequalities and Solving Them

Inequalities are mathematical expressions that show the relative size or order of two values. They are denoted by symbols like \( > \), \( < \), \( \geq \) (greater than or equal to), and \( \leq \) (less than or equal to). In our exercise, the inequality in question is \( x^{2}<10 \) which we solve to find numbers that make the inequality true.

When tackling inequalities, it's useful to think about them in the context of possible solutions — similar to solving an equation, except the answer is typically a range or set of numbers rather than a single value. Here, we discovered that 1, 2, and 3 satisfy the inequality, as squaring any of these natural numbers results in a number less than 10.