Sets and Numbers Overview

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Questions and Answers

Which of the following represents the set of natural numbers?

{x + iy : x, y are real numbers and i = -1}

{... -3, -2, -1, 0, 1, 2, ...}

{1, 2, 3, ... } (correct)

{a/b : a, b are integers and b 0}

Which of the following represents the set of integers?

{a/b : a, b are integers and b 0}

{1, 2, 3, ... }

{x + iy : x, y are real numbers and i = -1}

{... -3, -2, -1, 0, 1, 2, ...} (correct)

Which of the following represents the set of irrational numbers?

Numbers that cannot be expressed as a fraction (correct)

All positive whole numbers

Numbers that can be expressed as a fraction

All negative whole numbers

Which of the following represents the set of complex numbers?

{x + iy : x, y are real numbers and i = -1} Signup and view all the answers

The set of natural numbers is a subset of the set of integers.

True Signup and view all the answers

The set of rational numbers is a subset of the set of irrational numbers.

False Signup and view all the answers

The set of real numbers is a subset of the set of complex numbers.

True Signup and view all the answers

What is the relationship between the sets N, Z, Q, R, and C?

N Z Q R C Signup and view all the answers

What is the definition of a function?

A relation f: X Y is called a function if and only if for each element x X, there exists a unique element y Y such that y = f(x). Signup and view all the answers

What is the domain of a function?

The set of all possible input values (x-values) of a function. Signup and view all the answers

What is the range of a function?

The set of all possible output values (y-values) of a function. Signup and view all the answers

What is the domain of the function f(x) = 2?

R Signup and view all the answers

What is the range of the function f(x) = 2?

{2} Signup and view all the answers

What is the domain of the function f(x) = x - 4 / x + 2?

R{-2} Signup and view all the answers

What is the range of the function f(x) = x - 4 / x + 2?

R{-2} Signup and view all the answers

What is the domain of the function f(x) = (x - 2)(x + 2)?

R Signup and view all the answers

What is the range of the function f(x) = (x - 2)(x + 2)?

R Signup and view all the answers

What is the domain of the function f(x) = (2x - 6)?

[3, ) Signup and view all the answers

What is the domain of the function f(x) = (2x + 1) / ((x + 1)(x - 4))?

R{-1, 4} Signup and view all the answers

How does a function establish a set of ordered pairs?

A function f establishes a set of ordered pairs (x, y) where x is an element in the domain of f and y is the corresponding output (f(x)). Signup and view all the answers

Study Notes

Sets and Numbers

Natural Numbers (N): The set of positive integers: {1, 2, 3,...}
Integers (Z): The set of whole numbers, including zero and negative numbers: {...-3, -2, -1, 0, 1, 2, 3,...}
Rational Numbers (Q): Numbers that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0. Examples: 1/2, -3/4, 5, 0.
Irrational Numbers (I): Numbers that cannot be expressed as a fraction of integers. Examples: √2, π.
Real Numbers (R): The set of all rational and irrational numbers. (R = Q ∪ I)
Complex Numbers (C): Numbers of the form x + yi, where x and y are real numbers, and i = √-1. (C includes all real and imaginary numbers).
Relationship between sets: Natural numbers are a subset of integers, integers are a subset of rational numbers, and rational numbers are a subset of real numbers; real numbers are a subset of complex numbers. (N ⊂ Z ⊂ Q ⊂ R ⊂ C)

Functions

Definition: A relation from a set X to a set Y is a function if each element in X corresponds to exactly one element in Y.
Domain (D_f or Dom(f)): The set of all possible input values (x-values) for a function.
Range (R_f or Ran(f)): The set of all possible output values (y-values) for a function.
Example function types : Square root, absolute value, quadratic (x²), cubic (x³).

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Description

Explore the various types of number sets in mathematics, including natural, integers, rational, irrational, real, and complex numbers. Discover the relationships between these sets and understand their importance in mathematical theory. This quiz will help solidify your understanding of these foundational concepts.