MA1502 GE1359 2021A Ch1 notes PDF

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Semester A, 2020-21

MA1502 / GE1359

Algebra

Dr. Emily Chan

Semester A, 2020-21

Chapter 1

Page 1

MA1502 / GE1359

Algebra

Chapter 1

Chapter 1: Sets and functions 1

2

Set Theory 1.1

Sets and elements

1.2

Set operations

1.3

Intervals

Functions 2.1

One-to-one functions

2.2

Onto functions

2.3

Bijective functions

2.4

Operations of functions

2.5

Composition of functions

2.6

Inverse functions

2.7

Examples of elementary functions

2.8

Even and odd functions & periodic functions

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2.9

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Chapter 1

Transformation of functions

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Chapter 1

1 SET THEORY 1.1

Sets and elements

Definition A set is a collection of distinct objects. Each object in a set is called an element or a member of that set. A set may contain a finite number of elements, infinitely many elements, or even no elements. For example, x ܸ ൌ ሼǡ ǡ ǡ ǡ ሽ is the set of all vowels of the English alphabets.

x  ൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ͷǡ ͸ǡ ͹ǡ ͺǡ ͻǡ ͳͲሽ is the set of all integers from 1 to 10. x  ൌ ሼʹǡ Ͷǡ ͸ǡ ͺǡ ͳͲǡ ǥ ሽ is the set of all positive even numbers.

They are all sets, and their elements are listed inside the curly brackets “{ }”. A set is a collection, but not a list. The order in which the elements are written is not important. For example,  ൌ ሼܽǡ ܾǡ ܿ ሽ ൌ ሼܾǡ ܽǡ ܿሽ ൌ ሼܿǡ ܾǡ ܽሽ. Dr. Emily Chan

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We specify a particular set either by listing its elements or by stating properties which characterize the elements of the set. In general, we use the notation ሼ‫ݔ‬ȁ‫ݔ‬ሽ to denote a set of objects that share some common properties. The vertical line “|” means “such that”. We may also write a “:” instead of the vertical line “|”.

For example, x ‫ ܥ‬ൌ ሼ‫ݔ‬ȁ‫ݔ‬Ͳ ൏ ‫ ݔ‬൏ ͳͲሽ means that ‫ ܥ‬is the set of odd numbers greater than 0 but less than 10. It can also be written as ‫ ܥ‬ൌ ሼͳǡ ͵ǡ ͷǡ ͹ǡ ͻሽ.

x

‫ ܦ‬ൌ ሼ‫ݔ‬ȁ‫ݔ‬‫ݔ‬ͷሽ means that ‫ ܦ‬is the set of

negative integers which are multiples of 5, i.e. ‫ ܦ‬ൌ ሼെͷǡ െͳͲǡ െͳͷǡ െʹͲǡ െʹͷǡ ǥ ሽ

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Algebra

Chapter 1

Venn diagram A set can be represented using Venn diagram. For example, a Venn diagram for the set ‫ ܥ‬ൌ ሼͳǡ ͵ǡ ͷǡ ͹ǡ ͻሽ is shown on the right.

Some notations: x

“‫ ”א‬means “belongs to” or “is an element of”. If “ܽ belongs to ܵ” or “ܽ is an element of ܵ”, we write ܽ ‫ܵ א‬.

x

“‫ ”ב‬means “does not belong to” or “is not an element of”. If “ܾ does not belong to ܵ” or Dzܾܵdz, we write ܾ ‫ܵ ב‬.

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Algebra

Chapter 1

“‫ ”ك‬means “is a subset of”.

If every element in set ‫ ܣ‬also belongs to set ‫ܤ‬, i.e. if ‫ ܣ א ݌‬implies ‫ܤ א ݌‬, then we

say that “‫ ܣ‬is a subset of ‫ ”ܤ‬or “‫ ܣ‬is contained in ‫ ”ܤ‬and we write ‫ܤ ك ܣ‬. x

“‫ ”م‬means “is not a subset of”. If there is at least one element which belongs to set ‫ ܣ‬but does not belong to set ‫ܤ‬,

we say that “‫ ܣ‬is not a subset of ‫ ”ܤ‬and we write ‫ܤ م ܣ‬. x

“‫ ”ؿ‬means “is a proper subset of”. If ‫ ܣ‬is a subset of ‫ܤ‬, but ‫ ܣ‬is not equal to ‫( ܤ‬i.e. there exists at least one element of

‫ ܤ‬which does not belong to ‫)ܣ‬, then we say that ‫ ܣ‬is a proper subset of ‫ܤ‬, denoted

by ‫ܤ ؿ ܣ‬.

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Semester A, 2020-21

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“‫ ”ف‬means “is not a proper subset of”. We write ‫ ܤ ف ܣ‬if ‫ ܣ‬is not a proper subset of ‫ܤ‬.

Remarks: 1. If ‫ܤ ك ܣ‬, then ‫ ܣ‬may or may not be equal to ‫ܤ‬, but if ‫ܤ ؿ ܣ‬, then ‫ ܣ‬is definitely not equal to ‫ܤ‬.

2. By the definition of a subset, any set is a subset of itself, i.e. ‫ ܣ ك ܣ‬but ‫ܣ ف ܣ‬. 3. Two sets are equal if each is contained in the other; that is, ‫ ܣ‬ൌ ‫ ܤ‬if and only if ‫ ܤ ك ܣ‬and ‫ܣ ك ܤ‬.

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Example 1 Given the sets ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ͷǡ ͸ǡ ͹ǡ ͺǡ ͻǡ ͳͲሽ, ‫ ܥ‬ൌ ሼͳǡ ͵ǡ ͷǡ ͹ǡ ͻሽ and ‫ ܧ‬ൌ ሼʹǡ Ͷǡ ͳͳሽ. Then

¾ ‫ ܥ‬is a subset of ‫ܣ‬, denoted by ࡯ ‫( ࡭ ك‬or we may use ‫)ܣ ؿ ܥ‬, since every element in ‫ܥ‬ also belongs to ‫ܣ‬.

¾ ‫ ܧ‬is not a subset of ‫ܣ‬, because ͳͳ ‫ ܧ א‬but ͳͳ ‫ܣ ב‬. We write ࡱ ‫࡭ م‬.

‫ܣكܥ‬

‫ܣمܧ‬

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Algebra

Chapter 1

Unless otherwise stated, all sets under investigation are assumed to be subsets of some fixed sets called the universal set and denoted by ܷ. We also use ‫ ׎‬to denote the empty or null set, i.e. the set which contains no elements; this set is regarded as a subset of every other set. Thus for any set ‫ܣ‬, we have ‫ܷ ك ܣ ك ׎‬. Example 2 In human population studies, the universal set consists of all the people in the world.

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Some commonly used sets in Mathematics include: ‫ ׎‬ൌ ሼሽ is called an “empty set” or “null set”, which contains no elements.

Գ ൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ǥ ሽ is the set of all natural numbers (positive integers).

Ժ ൌ ሼͲǡ േͳǡ േʹǡ േ͵ǡ ǥ ሽ is the set of all integers. ௣

Է ൌ ቄ ȁ‫݌‬ǡ ‫ א ݍ‬Ժ‫Ͳ ് ݍ‬ቅ = the set of all rational numbers. ௤

Թ ൌ the set of all real numbers.

ԧ ൌ ሼܽ ൅ ܾ݅ȁܽǡ ܾ ‫ א‬Թሽ ൌthe set of all complex numbers. Using notations of subsets, we have ‫ ؿ ׎‬Գ ‫ ؿ‬Ժ ‫ ؿ‬Է ‫ ؿ‬Թ ‫ ؿ‬ԧǤ

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Example 3 Prove the following statement: “If ‫ ܣ‬is a subset of the empty set ‫׎‬, then ‫ ܣ‬ൌ ‫׎‬.” Solution The null set ‫ ׎‬is a subset of every set; in particular, ‫ܣ ك ׎‬. But, by hypothesis, ‫; ׎ ك ܣ‬ hence ‫ ܣ‬ൌ ‫׎‬.

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Theorem Let ‫ܣ‬, , ‫ ܥ‬be any sets. Then (a) ‫;ܣ ك ܣ‬

(b) if ‫ ܤ ك ܣ‬and ‫ ܣ ك ܤ‬, then ‫ ܣ‬ൌ ‫ ;ܤ‬and

(c) if ‫ ܤ ك ܣ‬and ‫ܥ ك ܤ‬, then ‫ܥ ك ܣ‬. Proof: (a) and (b) follows from the definition.

To proof (c), we must show that each element in ‫ ܣ‬also belongs to ‫ܥ‬.

Let ‫ܣ א ݔ‬.

Now ‫ ܤ ك ܣ‬implies ‫ܤ א ݔ‬.

But ‫ܥ ك ܤ‬, hence ‫ܥ א ݔ‬.

We have shown that ‫ ܣ א ݔ‬implies ‫ܥ א ݔ‬, therefore ‫ܥ ك ܣ‬.

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Semester A, 2020-21

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Example 4 Consider the following sets of figures in the Euclidean plane: ‫ ܣ‬ൌ ሼ‫ݔ‬ȁ‫ݔ‬ሽ, ‫ ܥ‬ൌ ሼ‫ݔ‬ȁ‫ݔ‬ሽ,

‫ ܤ‬ൌ ሼ‫ݔ‬ȁ‫ݔ‬ሽ,

‫ ܦ‬ൌ ሼ‫ݔ‬ȁ‫ݔ‬ሽ.

Determine which sets are proper subsets of any of the others. Solution Since a square has 4 right angles it is a rectangle, since it has 4 equal sides it is a rhombus, and since it has 4 sides it is a quadrilateral. Thus, ‫ܣ ؿ ܦ‬, ‫ ܤ ؿ ܦ‬and ‫ܥ ؿ ܦ‬.

That is, ‫ ܦ‬is a subset of the other three.

Also, since there are examples of rectangles, rhombuses and quadrilaterals which are not squares, ‫ ܦ‬is a proper subset of the other three.

In a similar manner, we see that ‫ ܤ‬is a proper subset of ‫ܣ‬, and ‫ ܥ‬is a proper subset of ‫ ܣ‬, i.e. ‫ ܣ ؿ ܤ‬and ‫ܣ ؿ ܥ‬. There are no other relations among the sets.

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Example 5 Determine which of the following sets are equal: ‫׎‬,

ሼ Ͳሽ,

ሼ ‫ ׎‬ሽ.

Solution Each is different from the other. The set ሼͲሽ contains one element, the number zero. The set ‫ ׎‬contains no elements; it is the empty set.

The set ሼ‫׎‬ሽ also contains one element, the empty set. ฀

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1.2

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Set operations

Given two arbitrary sets ‫ ܣ‬and ‫ܤ‬. We can combine the two sets to form new sets by using set operations: x Union The union of ࡭ and ࡮, denoted by ࡭ ‫࡮ ׫‬, is the set of elements which belong to either

࡭ or ࡮ or both of them. That is, ‫ ܤ ׫ ܣ‬ൌ ሼ‫ݔ‬ȁ‫ܣ א ݔ‬‫ܤ א ݔ‬ሽ. Here, “or” is used in the sense of both and/or.

E.g. If ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ሽ and ‫ ܤ‬ൌ ሼʹǡ ͵ǡ Ͷሽ, then ‫ ܤ ׫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ǡ Ͷሽ.

E.g. If ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ሽ and ‫ ܥ‬ൌ ሼͶǡ ͸ሽ, then ‫ ܥ ׫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ͸ሽ.

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x Intersection The intersection of ࡭ and ࡮, denoted by ࡭ ‫࡮ ת‬, is the set of elements which belong to

both ࡭ and ࡮. That is, ‫ ܤ ת ܣ‬ൌ ሼ‫ݔ‬ȁ‫ܣ א ݔ‬‫ܤ א ݔ‬ሽ.

E.g. If ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ሽ and ‫ ܤ‬ൌ ሼʹǡ ͵ǡ Ͷሽ, then ‫ ܤ ת ܣ‬ൌ ሼʹǡ ͵ሽ.

E.g. If ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ሽ and ‫ ܥ‬ൌ ሼͶǡ ͸ሽ, then ‫ ܥ ת ܣ‬ൌ ‫׎‬. That is, ‫ ܣ‬and ‫ ܥ‬do not have any

elements in common. Then ‫ ܣ‬and ‫ ܥ‬are said to be disjoint.

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x Complement The relative complement of ࡭ with respect to ࡮, denoted by ࡮࡭, is the set of elements which belong to ࡮ but not to ࡭.

That is, ‫ܤ‬‫ ܣ‬ൌ ሼ‫ݔ‬ȁ‫ܤ א ݔ‬ǡ ‫ܣ ב ݔ‬ሽ. The line “\” means “exclude”.

E.g. If ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ሽ and ‫ ܤ‬ൌ ሼʹǡ ͵ǡ Ͷሽ, then ‫ܤ‬‫ ܣ‬ൌ ሼͶሽ and ‫ܣ‬‫ ܤ‬ൌ ሼͳሽ.

E.g. Թሼͳǡ ͵ሽ is the set of all real numbers except 1 and 3. Observe that ‫ܤ‬‫ ܣ‬and ‫ ܣ‬are disjoint, i.e. ሺ‫ܤ‬‫ܣ‬ሻ ‫ ܣ ת‬ൌ ‫׎‬. Dr. Emily Chan

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The absolute complement or, simply, complement of ‫ܣ‬, denoted by ‫ܣ‬௖ , is the set of

elements which do not belong to ࡭. That is,

‫ܣ‬௖ ൌ ሼ‫ݔ‬ȁ‫ܷ א ݔ‬ǡ‫ܣ ב ݔ‬ሽǤ

‫ܣ‬௖ is the set which consists of the elements in ܷ that are not in ‫ܣ‬.

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Example 6 Let ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ǡ Ͷሽ and ‫ ܤ‬ൌ ሼ͵ǡ Ͷǡ ͷǡ ͸ሽ , where ܷ ൌ ሼͳǡ ʹǡ ͵ǡ ǥ ሽ is the set of all positive integers. Write down the following sets: (a) ‫ܤ ׫ ܣ‬

(b) ‫ܤ ת ܣ‬

(c) ‫ܣ‬‫ܤ‬

(d) ‫ܣ‬௖

Solution (a) ‫ ܤ ׫ ܣ‬ൌ (b) ‫ ܤ ת ܣ‬ൌ (c) ‫ܣ‬‫ ܤ‬ൌ (d) ‫ܣ‬௖ ൌ Dr. Emily Chan

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Example 7 Let ܷ ൌ ሼͳǡ ʹǡ ǥ ǡ ͺǡ ͻሽ, ‫ ܣ‬ൌ ሼͳǡ ʹǡ ͵ǡ Ͷሽ, ‫ ܤ‬ൌ ሼʹǡ Ͷǡ ͸ǡ ͺሽ and ‫ ܥ‬ൌ ሼ͵ǡ Ͷǡ ͷǡ ͸ሽ. Determine the following sets: (a) ‫ܣ‬௖

(c) ሺ‫ܥ ת ܣ‬ሻ௖

(b) ‫ܥ ת ܣ‬

(d) ‫ܤ ׫ ܣ‬

(e) ‫ܤ‬‫ܥ‬

Solution (a) ‫ܣ‬௖ consists of the elements in ܷ that are not in ‫ܣ‬. Hence, ‫ܣ‬௖ ൌ

(b) ‫ ܥ ת ܣ‬consists of the elements in both ‫ ܣ‬and ‫ܥ‬. Hence, ‫ܥתܣ‬ൌ

(c) ሺ‫ ܥ ת ܣ‬ሻ௖ consists of the elements in ܷ that are not in ‫ܥ ת ܣ‬. Hence, ሺ‫ܥ ת ܣ‬ሻ௖ ൌ

(d) ‫ ܤ ׫ ܣ‬consists of the elements in ‫ ܣ‬or ‫( ܤ‬or both). Hence, ‫ܤ׫ܣ‬ൌ

(e) ‫ܤ‬‫ ܥ‬consists of the elements in ‫ ܤ‬which are not in ‫ܥ‬. Hence, ‫ܤ‬‫ ܥ‬ൌ Dr. Emily Chan

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Sets under the above operations satisfy various laws or identities which are listed in the table below (Table 1.1). In fact we state the following theorem:

Theorem: Sets satisfy the laws in Table 1.1.

Table 1.1

Laws of the algebra of sets Idempotent Laws

1a. ‫ ܣ ׫ ܣ‬ൌ ‫ܣ‬

1b. ‫ ܣ ת ܣ‬ൌ ‫ܣ‬ Associative Laws

2a. ሺ‫ ܤ ׫ ܣ‬ሻ ‫ ܥ ׫‬ൌ ‫ ׫ ܣ‬ሺ‫ ܥ ׫ ܤ‬ሻ

ሺ‫ܤ ת ܣ‬ሻ ‫ ܥ ת‬ൌ ‫ ת ܣ‬ሺ‫ ܥ ת ܤ‬ሻ

2b.

Commutative Laws 3a. ‫ ܤ ׫ ܣ‬ൌ ‫ܣ ׫ ܤ‬

3b.

‫ܤתܣ‬ൌ‫ܣתܤ‬

Distributive Laws 4a. ‫ ׫ ܣ‬ሺ‫ ܥ ת ܤ‬ሻ ൌ ሺ‫ܤ ׫ ܣ‬ሻ ‫ ת‬ሺ‫ ܥ ׫ ܣ‬ሻ Dr. Emily Chan

4b.

‫ ת ܣ‬ሺ‫ ܥ ׫ ܤ‬ሻ ൌ ሺ‫ܤ ת ܣ‬ሻ ‫ ׫‬ሺ‫ܥ ת ܣ‬ሻ Page 22

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Identity Laws 5a. ‫ ׎ ׫ ܣ‬ൌ ‫ܣ‬

5b. ‫ ܷ ת ܣ‬ൌ ‫ܣ‬

6a. ‫ ܷ ׫ ܣ‬ൌ ܷ

6b. ‫ ׎ ת ܣ‬ൌ ‫׎‬

Complement Laws 7a. ‫ܣ ׫ ܣ‬௖ ൌ ܷ

7b.

8a. ሺ‫ܣ‬௖ ሻ௖ ൌ ‫ܣ‬

‫ܣ ת ܣ‬௖ ൌ ‫׎‬

8b. ܷ ௖ ൌ ‫ ׎‬, ‫׎‬௖ ൌ ܷ De Morgan’s Laws

9a. ሺ‫ ܤ ׫ ܣ‬ሻ௖ ൌ ‫ܣ‬௖ ‫ܤ ת‬௖

9b.

ሺ‫ܤ ת ܣ‬ሻ௖ ൌ ‫ܣ‬௖ ‫ܤ ׫‬௖

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Example 8 (For your reference) Prove the following statements: (a) ‫ ܤ ׫ ܣ‬ൌ ሺ‫ܣ‬௖ ‫ ܤ ת‬௖ ሻ௖

(b)

‫ܣ‬‫ ܤ‬ൌ ‫ܤ ת ܣ‬௖ .

Solution (a) ሺ‫ܣ‬௖ ‫ ܤ ת‬௖ ሻ௖ ൌ ሼ‫ݔ‬ȁ‫ܣ ב ݔ‬௖ ‫ܤ ת‬௖ ሽ

ൌ ሼ‫ݔ‬ȁ‫ ܣ ב ݔ‬௖ ‫ܤ ב ݔ‬௖ ሽ

ൌ ሼ‫ݔ‬ȁ‫ܣ א ݔ‬‫ܤ א ݔ‬ሽ ൌ ሼ‫ݔ‬ȁ‫ܤ ׫ ܣ א ݔ‬ሽ

ൌ ‫ܤ׫ܣ‬

(b) ‫ܣ‬‫ ܤ‬ൌ ሼ‫ݔ‬ȁ‫ܣ א ݔ‬‫ܤ ב ݔ‬ሽ

ൌ ሼ‫ݔ‬ȁ‫ܣ א ݔ‬‫ܤ א ݔ‬௖ ሽ

ൌ ‫ܤ ת ܣ‬௖ Dr. Emily Chan

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Theorem: (For your reference) Each of the following conditions is equivalent to ‫ܤ ك ܣ‬: (a) ‫ ܤ ת ܣ‬ൌ ‫ܣ‬

(b) ‫ ܤ ׫ ܣ‬ൌ ‫ܤ‬

(c) ‫ܤ‬௖ ‫ܣ ك‬௖

(d) ‫ܤ ת ܣ‬௖ ൌ ‫׎‬ (e) ‫ܣ ׫ ܤ‬௖ ൌ ܷ

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Semester A, 2020-21

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Algebra

Chapter 1

Intervals

Recall that Թ is the set of all real numbers. Let ܽ and ܾ be two distinct real numbers where

ܽ ൏ ܾ. We use the following notations to describe some subsets of real numbers (known as intervals): ሺܽǡ ܾሻ ൌ ሼ‫ א ݔ‬Թȁܽ ൏ ‫ ݔ‬൏ ܾሽ ሾܽǡ ܾሻ ൌ ሼ‫ א ݔ‬Թ ȁܽ ൑ ‫ ݔ‬൏ ܾሽ

ሺܽǡ ܾሿ ൌ ሼ‫ א ݔ‬Թȁܽ ൏ ‫ ݔ‬൑ ܾሽ

ሾܽǡ ܾሿ ൌ ሼ‫ א ݔ‬Թ ȁܽ ൑ ‫ ݔ‬൑ ܾሽ ሾܽǡ λሻ ൌ ሼ‫ א ݔ‬Թȁ‫ ݔ‬൒ ܽሽ

ሺܽǡ λሻ ൌ ሼ‫ א ݔ‬Թȁ‫ ݔ‬൐ ܽሽ

ሺെλǡ ܽሿ ൌ ሼ‫ א ݔ‬Թ ȁ‫ ݔ‬൑ ܽሽ

ሺെλǡ ܽሻ ൌ ሼ‫ א ݔ‬Թȁ‫ ݔ‬൏ ܽሽ Թ ൌ ሺെλǡ λሻ

Note: Never write ሾܽǡ λሿ, ሺܽǡ λሿ, ሾെλǡ ܽሿ and ሾെλǡ ܽሻ. Dr. Emily Chan

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Algebra

Chapter 1

2 FUNCTIONS ¾ A function is a rule that assigns a unique value ݂ሺ‫ݔ‬ሻ to any ‫ ݔ‬from a set called the domain. ¾ The domain of a function is the set of all possible input values (i.e. all possible values of ‫) ݔ‬ for which the function is defined. ¾ The codomain of a function is the set which contains all possible output values. ¾ The range is the set of all output values (i.e. all values of ‫ ݕ‬or ݂ሺ‫ݔ‬ሻ), which actually result from using the function formula. ¾ In general, the range of a function is a subset of its codomain but not necessarily the same set.

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Semester A, 2020-21

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MA1502 / GE1359

Algebra

Chapter 1

¾ Clearly, the range of a function depends on what you put into the function (domain) and the function itself. ¾ If set ‫ ܣ‬is the domain of ݂ and set ‫ ܤ‬is the codomain of ݂, we write ݂ǣ ‫ ܣ‬՜ ‫ܤ‬,

which reads “݂ is a function of ‫ ܣ‬into ‫”ܤ‬.

Further, if ܽ ‫ ܣ א‬then the element in ‫ ܤ‬which is assigned to ܽ is called the image of ܽ

under ݂ and is denoted by which reads “݂ of ܽ”.

݂ሺܽሻ

For example, we may write the following to define a function: Let ݂ǣ Թ ՜ Թ be defined by

Dr. Emily Chan

݂ ሺ‫ ݔ‬ሻ ൌ ‫ ݔ‬ଶ ൅ ‫ ݔ‬൅ ͳ. Page 28

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Algebra

Chapter 1

¾ If ‫ ܣ א ݔ‬and ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ‫( ܤ א‬for example, ‫ ݕ‬ൌ ‫ ݔ‬ଶ ൅ ‫ ݔ‬൅ ͳ), then ‫ ݔ‬is called the independent variable and ‫ ݕ‬is called the dependent variable.

Example 9 Let ‫ ܣ‬ൌ ሼܽǡ ܾǡ ܿǡ ݀ሽ and ‫ ܤ‬ൌ ሼܽǡ ܾǡ ܿ ሽ . Define a function ݂ of ‫ ܣ‬into ‫ ܤ‬by the

correspondence ݂ሺܽሻ ൌ ܾ, ݂ሺܾሻ ൌ ܿ, ݂ሺܿሻ ൌ ܿ and ݂ሺ݀ሻ ൌ ܾ. By this definition, the image, for example, of ܾ is ܿ.

Example 10 Let ‫ ܣ‬ൌ ሼെͳǡ ͳሽ. Let ݂ assign to each rational number in Թ the number 1, and to each

irrational number in Թ the number െͳ. Then ݂ǣ Թ ՜ ‫ܣ‬, and ݂ can be defined concisely by ͳ‫ݔ‬ Ǥ ݂ሺ‫ݔ‬ሻ ൌ ቄ െͳ‫ݔ‬

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Algebra

Chapter 1

Example 11 Let ‫ ܣ‬ൌ ሼܽǡ ܾǡ ܿǡ ݀ሽ and ‫ ܤ‬ൌ ሼ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሽ. Let ݂ǣ ‫ ܣ‬՜ ‫ ܤ‬be defined by the diagram

ܽ ܾ ܿ

‫ݔ‬

‫ݕ‬ ‫ݖ‬

݀

Notice that the functions in Example 10 or on page 28 are defined by specific formulas. But this need not always be the case, as is indicated by the other examples. The rules of correspondence which define functions can be diagrams as in Example 11, or when the domain is finite, the correspondence can be listed for each element in the domain as in Example 9. Dr. Emily Chan

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¾

MA1502 / GE1359

Algebra

Chapter 1

If ‫ ܣ‬and ‫ ܤ‬are sets, then a function ݂ of ‫ ܣ‬into ‫ ܤ‬is frequently called a mapping of

‫ ܣ‬into ‫ ;ܤ‬and the notation

݂ǣ ‫ ܣ‬՜ ‫ܤ‬

is then read “݂ maps ‫ ܣ‬into ‫”ܤ‬. We can also denote a mapping, or function, ݂ of ‫ܣ‬

into ‫ ܤ‬by

௙

‫ܣ‬՜‫ܤ‬ or the diagram ݂

‫ܤ‬

‫ܣ‬

¾

If ݂ and ݃ are functions defined on the same domain ‫ ܦ‬and if ݂ሺܽሻ ൌ ݃ሺܽሻ for every

ܽ ‫ܦ א‬, then the functions ݂ and ݃ are equal and we write ݂ ൌ ݃.

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