The symbol ℝ represents the set of all real numbers, which includes all positive and negative integers, fractions, and irrational numbers. It is a fundamental concept in mathematics that is essential for understanding the domain and range of functions.
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The set of real numbers, ℝ, is an infinite set that extends from negative infinity to positive infinity.
Real numbers can be expressed as decimal representations, both terminating and non-terminating (irrational numbers).
The set of real numbers, ℝ, is dense, meaning that between any two real numbers, there are infinitely many other real numbers.
Real numbers are used to represent quantities that can have fractional or decimal parts, such as length, weight, and time.
The set of real numbers, ℝ, is the foundation for understanding the domain and range of functions in mathematics.
Review Questions
Explain how the set of real numbers, ℝ, is related to the domain and range of functions.
The set of real numbers, ℝ, is the foundation for understanding the domain and range of functions. The domain of a function represents the set of all possible input values, which are typically real numbers. Similarly, the range of a function represents the set of all possible output values, which are also typically real numbers. The properties of the real number system, such as its density and infinite nature, have important implications for the behavior and properties of functions.
Describe the relationship between the set of integers and the set of real numbers, ℝ.
The set of integers is a subset of the set of real numbers, ℝ. The integers include all whole numbers, both positive and negative, including zero. The real numbers, ℝ, extend beyond the integers to include all fractions and irrational numbers. This means that every integer is a real number, but not every real number is an integer. Understanding the relationship between these two sets is crucial for determining the domain and range of functions, as the domain and range may be restricted to specific subsets of the real numbers, such as the integers.
Analyze the significance of the density of the real number system, ℝ, in the context of domain and range.
The density of the real number system, ℝ, means that between any two real numbers, there are infinitely many other real numbers. This has important implications for the domain and range of functions. Since the real numbers are dense, the domain and range of a function can potentially include an infinite number of values, allowing for a wide variety of possible input and output values. This property of the real number system enables the study of continuous functions, which are central to the understanding of domain and range in pre-calculus and beyond.