0.10 Systems of Equations
1.1.1 Introduction
Idea of definition Asetis a collection of objects.
This is not a definition because we have not defined what acollectionis. If we give a definition forcollec- tion, it must involve something that have not been defined. It is impossible to define everything. In mathematics, setis a fundamental concept that cannot be defined. The idea of definition given above describes what a set is using daily language. This helps us “understand” the meaning of a set.
Terminology An object in a set is called anelementor amemberof the set.
To describe sets, we can uselistingordescription.
[Listing] To denote a set with finitely many elements, we can list all the elements of the set and enclose them by braces. For example,
{1,2,3}
is the set which has exactly three elements, namely 1, 2 and 3.
If we want to denote the set whose elements are the first one hundred positive integers, it is impractical to write down all the elements. Instead, we write
{1,2,3, . . . ,99,100}, or simply {1,2, . . . ,100}.
The three dots “. . .”(read “and so on”) means that the pattern is repeated, up to the number(s) listed at the end.
Suppose in a problem, we consider a set, say{1,2, . . . ,100}. We may have to refer to the set later many times. Instead of writing{1,2, . . . ,100}repeatedly, we can give it a name by using a symbol to represent the set. Usually, we use small letters (eg.a,b, . . .) to denote objects and capital letters (eg.A,B, . . .) to denote sets.
For example, we may write
• “LetA={1,2, . . . ,100}.”
24 Chapter 1. Sets, Real Numbers and Inequalities
which means that the set{1,2, . . . ,100}is given the “name”A. If we want to refer to the set later, we can just writeA. For example,
• “LetA={1,2, . . . ,100}. Then 100 is an element ofA, but 101 is not an element ofA.”
If we consider another set, say{1,2,3,4,5}and want to give it a name, we must not use the symbolAagain, because in the problem,Aalways means the set{1,2, . . . ,100}. For example,
• “Let A = {1,2, . . . ,100}. Let B = {1,2,3,4,5}. Then every element ofBis also an element of A. But there are elements ofAthat are not elements ofB.”
Remark The equality sign “=” can be used in several ways as the following examples illustrate.
(1) 1+2=3.
(2) x2+1=5.
(3) LetA={1,2,3}.
The equality sign in (1) means equality of two quantities: the quantity on the left and the quantity on the right are equal.
The equality sign in (2) is an equality in an equation. It is true whenx =2 (for example) and it is not true when x=1 (for example). Instead of using the equality sign, some authors use “==”. The equation in (2) may be written as
(20) x2+1==5.
The equality sign in (3) has a different meaning. The sentence in (3) means that the set{1,2,3}is denoted by A. The symbol “=” assigns a name to an object(a set is also an object). The name is written on the left side and the object on the right side. Instead of using the equality sign, some authors use the symbol “:=”. The sentence in (3) may be written as
(30) LetA:={1,2,3}.
In this course, we will not use the notations “:=” and “==”. Readers can determine the meaning of “=” from the context.
Notation Given an objectxand a setA, eitherxis an element ofAorxis not an element ofA.
(1) Ifxis an element ofA, we writex∈A(read “x belongs to A”).
(2) Ifxis not an element ofA, we writex<A(read “x does not belong to A”).
There is a set that has no element. It is called theempty set, denoted by∅. This is a Scandinavian letter, a zero 0 together with a slash/.
Definition The set that has no element is called theempty setand is denoted by∅.
RemarkBecause the empty set has no element, if we list all the elements of it and enclose “them” by braces, we get{ }. This is an alternative notation for the empty set.
[Description] Another way to denote a set is to describe a common property of the elements of the set, using the following notation:
{x:P(x)} or {x|P(x)}
1.1. Sets 25
read “the set of all x such that P(x) (is true)”. For example, the set whose elements are the first one hundred positive integers can be expressed as
(†) {x:xis a positive integer less than 101}
In considering “property”, it is understood that the property applies to a certain collection of objects only.
For example, when we say “an old person” (a person is said to be old if his or her age is 65 or above), the property of being “old” is applied to people. It is meaningless to say “this is an old atom” (unless we have a definition which tells whether an atom is old or not).
The property of being a positive integer less than 101 is applied to numbers. In this course, we consider real numbers only. The set of all real numbers is denoted byR. In considering the set given in (†), it is understood thatxis a real number. To make this explicit, we write
(‡) {x∈R:xis a positive integer less than 101}
read “the set of all x belonging toRsuch that x is a positive integer less than101”.
Notation
(1) The set of all real numbers is denoted byR.
(2) The set of all rational numbers is denoted byQ.
(3) The set of all integers is denoted byZ.
(4) The set of all positive integers is denoted byZ+. (5) The set of all natural numbers is denoted byN.
Definition
(1) Arational numberis a number that can be written in the form p
q wherepandqare integers andq,0.
(2) Positive integers together with 0 are callednatural numbers.
RemarkSome authors do not include 0 as natural number. In that case,Nmeans the set of all positive integers.
Example
(1) To say that 2 is a natural number, we may write 2∈N.
(2) To say that 2 is a rational number, we may write 2∈Q.
Note: The number 2 is a rational number because it can be written as 2 1or 6
3etc.
(3) To say thatπis not a rational number, we may writeπ<Q.
Note:π,22
7; the rational number 22
7 is only an approximation toπ.
Definition LetAandBbe sets. If every element ofAis also an element of Band vice versa, then we say that AandBareequal, denoted byA=B.
Remark
• In mathematics, definitions are important. Students who want to take more courses in mathematics must pay attention to definitions. Understand the meaning, give examples, give nonexamples.
26 Chapter 1. Sets, Real Numbers and Inequalities
• In the definition, the first sentence “Let A and B be sets” describes the setting. The definition for equality applies to setsonly and does not apply to other objects. Of course, we can consider equality of other objects, but it is another definition.
• In the first sentence “Let A and B be sets”, the use of plural “sets” does not mean thatAandBare two different sets. It also includes the case whereA andBare the same set. The following are alternative ways to say this:
♦ Let A and B be set(s).
♦ Let A be a set and let B be a set.
However, these alternative ways are rather cumbersome and will not be used in most situations.
• Some students may not be familiar with the use of the word “let”. It is used very often in mathematics.
Consider the following sentences:
♦ LetA={1,2,3,4,5}.
♦ LetAbe a set.
The word “let” appears in both sentences. However, the meanings of “let” in the two sentences are quite different. In the first sentence, “let” meansdenotewhereas in the second sentence, it meanssuppose. The definition for equality of sets can also be stated in the following ways:
♦ SupposeAandBare sets. If every element ofAis also an element ofBand vice versa, then we say thatAandBareequal.
♦ IfAandBare sets and if every element ofAis also an element ofBand vice versa, then we say that AandBareequal.
• The definition can also be stated in a way that the assumption thatAandBare sets is combined with the condition for equality ofAandB.
♦ If every element of a setAis also an element of a setBand vice versa, then we say thatAandBare equal.
• The definition tells that ifAandBare sets having the same elements, thenA= B. Conversely, it also tells that ifAandBare sets andA= B, thenAandBhave the same elements because this is the condition to check whetherAandBare equal. Some mathematicians give the definition using iff:
♦ Let Aand Bbe sets. We say that Aand Bare equalif and only if every element of A is also an element ofBand vice versa.
• Sometimes, we also give definition of a concept together with its “opposite”. The following is a definition of equality of sets together with its opposite. In this course, we will use the following format
(a) describe the setting;
(b) give condition(s) for the concept;
(c) give condition(s) for the opposite concept, whenever it is appropriate.
Definition LetAandBbe sets. If every element ofAis also an element ofBand vice versa, then we say that
1.1. Sets 27
AandBareequal, denoted byA=B. Otherwise, we say thatAandBareunequal, denoted byA, B.
Example LetA = {1,3,5,7,9}and let B = {x ∈Z : xis a positive odd number less than 10}. Then we have A=B, that is,AandBare equal. This is because every element ofAis also an element ofBand vice versa.
Recall:Zis the set of all integers. ThusBis the set of all integers that are positive, odd and less than 10.
RemarkTo prove that the setsAandBin the above example are equal, we check whether the condition given in the definition is satisfied. This is calledproof by definition.
Example LetA={1,3,5,7,9}and letB={x∈Z+:xis a prime number less than 10}. Then we haveA, B.
Recall:Z+is the set of all positive integers.
Proof The number 9 is an element ofA, but it is not an element ofB. Therefore, it is not true that every element
ofAis also an element ofB. Hence we haveA,B.
FAQ In the above two examples, the assertions are quite obvious. Do we need to prove them?
Answer Sometimes, mathematicians also write “obvious” in proofs of theorems. To some people, a result may be obvious; but, it may not be obvious to other people. If you say obvious, make sure that it is really obvious—
if your classmates ask you why, you should be able to explain to them.
It is impractical to explain everything. In proving theorems or giving solutions to examples, reasons that are “obvious” will not be given. When you answer questions, you should use your own judgment.
Remark Because it is impractical (in fact, impossible) to explain everything, discussion below will not be so detail as that above. If you don’t understand a concept, read the definition again. Try different ways to understand it. Relate it with what you have learnt. Guess what the meaning is. See whether your guess is correct if you apply it to examples. . .
Example LetA={1,2,3}and letB={1,3,2}. Then we haveA= B.
Proof Obvious (use definition).
The above example shows that in listing elements of a set, order is not important. It should also be noted that in listing elements, there is no need to repeat the elements. For example,{1,2,3,2,1}and{1,2,3}are the same set.
Definition LetAandBbe sets. If every element ofAis also an element ofB, then we say thatAis asubsetof B, denoted byA⊆B. Otherwise, we say thatAisnot a subsetofB, denoted byA* B.
Note
(1) A⊆A.
(2) A=Bif and only ifA⊆ BandB⊆ A.
(3) A*Bmeans that there is at least one element ofAthat is not an element ofB.
28 Chapter 1. Sets, Real Numbers and Inequalities
RemarkInstead ofA⊆B, some authors useA⊂ Bto denoteAis a subset ofB.
Example LetA={1,2,3,4,5},B={1,3,5}andC={2,4,6}.
Then we haveB⊆AandC*A.
The relation betweenA,BandCcan be described by the dia- gram shown in Figure 1.1.
1
2
3 6
4 5
Figure 1.1
FAQ For the given setsA,BandC, we also have the following:
(1) A* B (2) A*C (3) C* B (4) B*C
Why are they omitted?
Answer Good and correct observation. Given three sets, there are six ways to pair them up. The example just
illustrates the meaning of⊆and*.
1.1.2 Set Operations