Problem 13 - Fundamentals of Mathematics

Question

What numbers can replace \(x\) so that each statement is true? \(0 \leq x<5, x\) is a natural number.

Step-by-Step Solution

Answer

The numbers 0, 1, 2, 3, and 4 can replace \(x\).

Step 1: Understanding Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. In some mathematical contexts, natural numbers are considered to start from 0. In this exercise, we will consider both. We need to identify which natural numbers are within the specified range.

Step 2: Identifying the Range

The expression given is \(0 \leq x < 5\). This means \(x\) can start from 0 and must be less than 5. Therefore, the possible integer values for \(x\) satisfying this condition are: 0, 1, 2, 3, 4.

Step 3: Checking Natural Number Condition

We now filter out these values based on the condition that \(x\) is a natural number. Depending on whether your math convention includes 0 as a natural number or not, here we will assume it does. The numbers 0, 1, 2, 3, and 4 are natural numbers.

Step 4: Final Solution

Considering the range \(0 \leq x < 5\) and that \(x\) must be a natural number, the numbers that can replace \(x\) are: 0, 1, 2, 3, and 4.

Key Concepts

Integer Values

An integer is any whole number that can be positive, negative, or zero. Notably, integers do not include fractions or decimals. They are represented by the set of numbers: \(..., -3, -2, -1, 0, 1, 2, 3, ...\).
When dealing with natural numbers, which are a subset of integers, we only focus on positive values starting from 0 or 1, depending on the context. Since natural numbers are included in the range provided in the problem, we focus on positive whole numbers from the set of integers.
In the given exercise, integer values refer specifically to the natural numbers that satisfy the expression \(0 \leq x < 5\), which are:

• 0
• 1
• 2
• 3
• 4

These values fit both criteria, being natural numbers and lying within the specified mathematical range.

Mathematical Range

A mathematical range determines a set of values that satisfy an expression or condition defined by inequalities. In our case, the expression \(0 \leq x < 5\) indicates a range.
This range defines the smallest (\