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?

<p>{x + iy : x, y are real numbers and i = -1} (C)</p> Signup and view all the answers

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

<p>True (A)</p> Signup and view all the answers

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

<p>False (B)</p> Signup and view all the answers

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

<p>True (A)</p> Signup and view all the answers

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

<p>N Z Q R C</p> Signup and view all the answers

What is the definition of a function?

<p>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).</p> Signup and view all the answers

What is the domain of a function?

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

What is the range of a function?

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

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

<p>R</p> Signup and view all the answers

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

<p>{2}</p> Signup and view all the answers

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

<p>R{-2}</p> Signup and view all the answers

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

<p>R{-2}</p> Signup and view all the answers

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

<p>R</p> Signup and view all the answers

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

<p>R</p> Signup and view all the answers

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

<p>[3, )</p> Signup and view all the answers

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

<p>R{-1, 4}</p> Signup and view all the answers

How does a function establish a set of ordered pairs?

<p>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)).</p> Signup and view all the answers

Flashcards

Natural Numbers

A set of all positive whole numbers, starting at 1 and increasing without limit. These numbers are used for counting and ordering.

Integers

A set of all whole numbers, including both positive and negative integers, and zero. They represent both positive and negative quantities.

Rational Numbers

A set of numbers that can be expressed as a fraction of two integers, where the denominator is not zero. They represent whole numbers and parts of whole numbers.

Irrational Numbers

Numbers that cannot be expressed as a fraction of two integers. They cannot be represented as a terminating or repeating decimal.

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Real Numbers

The set of all rational and irrational numbers. They represent all possible numbers on a number line.

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Complex Numbers

A set of numbers that includes the imaginary unit 'i', which is the square root of -1. These numbers can be expressed as a sum of a real and an imaginary part.

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What is a set?

A collection of elements, where each element is unique and well-defined. Sets are denoted by curly braces.

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What is a subset?

A relationship between sets, where every element of one set is also an element of the other set. It is denoted by ⊆.

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What is a function?

A rule that associates each element in a set called the ‘domain’ with a unique element in another set called the ‘range’.

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What is the domain of a function?

The set of all possible input values for a function, represented by 'Df' or 'Dom(f).'

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What is the range of a function?

The set of all possible output values for a function, represented by 'Rf' or 'Ran(f).'

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What is a constant function?

A function that assigns a constant value to every input. It has a horizontal line as its graph.

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What is a quadratic function?

A function that involves the square of the input variable. Its graph is a U-shaped curve.

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What is a square root function?

A function that involves the square root of the input variable. Its graph is a curve that starts at the origin and increases slowly.

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What is an absolute value function?

A function that involves taking the absolute value of the input variable. Its graph is a V-shaped curve.

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What is a rational function?

A function that involves a rational expression, where both numerator and denominator are polynomials. Its graph can have vertical and horizontal asymptotes.

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What is the graph of a function?

The representation of a function using points plotted on a coordinate system based on ordered pairs (x,f(x)).

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How to find the domain of a function?

The process of finding the domain of a function by identifying when it is defined. It involves considering restrictions on the input values.

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How to find the range of a function?

The process of finding the range of a function by determining the set of all possible output values. It involves analyzing the function's behavior over its domain.

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How to sketch the graph of a function?

The process of sketching the graph of a function based on its definition and properties. It involves plotting points and connecting them to form a curve.

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How does a function relate to sets of ordered pairs?

The mathematical idea that a function can be represented by a set of ordered pairs (x,y), where x represents the input and y represents the output.

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What are the restrictions on the domain of a function?

The process of identifying the limitations or restrictions on the input values of a function. It involves considering factors such as square roots, fractions, and logarithms.

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How to determine the behavior of a function as x approaches infinity?

The process of determining the behavior of a function as the input values approach infinity or negative infinity. It involves analyzing the function's growth or decay patterns.

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How to find values that should be excluded from the domain?

The process of identifying any values of x that make the denominator of a rational function equal to zero. These values must be excluded fromthe domain.

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How to simplify a function?

The process of simplifying a given function by performing algebraic operations or using trigonometric identities. It involves expressing the function in a more compact or manageable form

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How to analyze a function?

The process of examining different aspects of a function, such as its domain, range, intervals of increasing and decreasing behavior, intercepts, and asymptotes. It involves understanding the function's properties and its graphical representation.

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How to evaluate a function?

The process of applying a function to a specific input value. It involves substituting the input value into the function's formula and evaluating the result.

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How to find the inverse of a function?

The process of finding the inverse of a function by swapping the roles of the input and output variables. It involves solving for the input variable in terms of the output variable.

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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, Ï€.
  • 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 (Df or Dom(f)): The set of all possible input values (x-values) for a function.
  • Range (Rf 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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