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Chapter Analysis
Intermediate16 pages • EnglishQuick Summary
This chapter introduces the concepts of relations and functions, starting with the Cartesian product of sets and the definition of relations. It describes functions as special types of relations that assign a unique output to each input. The chapter explains various types of functions such as real valued functions, polynomial functions, and others with examples and their properties. The chapter finishes with exercises that help reinforce these concepts.
Key Topics
- •Cartesian Product
- •Definition of Relation
- •Definition of Function
- •Types of Functions
- •Domain and Range
- •Real-Valued Functions
- •Polynomial Functions
- •Algebra of Functions
Learning Objectives
- ✓Understand the concept of Cartesian products and how to form them.
- ✓Differentiate between relations and functions.
- ✓Learn to identify and describe functions.
- ✓Understand domain and range of relations and functions.
- ✓Recognize different types of functions including polynomial and real-valued functions.
- ✓Understand basic operations on functions such as addition, subtraction, multiplication, and division.
Questions in Chapter
Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify your answer.
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Let A = {9,10,11,12,13} and let f : A→N be defined by f (n) = the highest prime factor of n. Find the range of f.
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Additional Practice Questions
What is a Cartesian product of two non-empty sets A and B?
easyAnswer: The Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
Explain the difference between a relation and a function.
mediumAnswer: A relation from a set A to a set B is a subset of the Cartesian product A × B. A function is a special type of relation where each element in set A is associated with exactly one element in set B.
How can you determine if a relation is a function?
mediumAnswer: A relation is a function if every element in the domain has a unique image in the codomain. There should be no repeated elements in the domain with different images.
What is the domain and range of the function f(x) = x^2?
easyAnswer: The domain of f(x) = x^2 is all real numbers R, and the range is all non-negative real numbers [0, ∞).
Define a real-valued function with an example.
easyAnswer: A real-valued function is a function whose domain is a subset of real numbers and the range is also a subset of real numbers. Example: f(x) = 2x + 1, x ∈ R.