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1 Ang aking kontrata: Ako, si, ay nangangakong magsisipag mag-aral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking bayang Pilipinas at para sa ikauunlad ng mundo.

2 THEOREMS (proposition, lemma, theorem, corollary which are deductively proven) AXIOMS (or postulates) Primitive/undefined terms Definitions Mathematics Division,IMSP,UPLB 2

3 Operations Mathematics Division,IMSP,UPLB 3

4 Basic Operations 1. Addition - denoted by + - result is called sum Ex = 5 Recall the rules for adding signed numbers = (-3) + (-5) = 8 + (-4) = 4 + (-8) = Mathematics Division,IMSP,UPLB 4

5 Basic Operations 2. Multiplication - denoted by x or, or just by juxtaposition - result is called product Ex. 2 x 3 = 6 Recall the rules for multiplying signed numbers 2 x 8 = (-3) x 6 = 3 x (-6) = (-4) x (-5) = Mathematics Division,IMSP,UPLB 5

6 Closure Property A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set. If an element outside the set is produced, then the operation is not closed. When you combine any two elements of the set, the result is also included in the set. Mathematics Division,IMSP,UPLB 6

7 Example If you add two real numbers, you will get another real number. Example: = 6.5 Since this process is always true, it is said that the the set of real numbers is closed under the operation of addition. There is simply no way to escape the set of real numbers when performing addition. Mathematics Division,IMSP,UPLB 7

8 Is the set of real numbers closed under multiplication? If you multiply two real numbers, you will get another real number. Example: 2(1/5)=2/5 Since this process is always true, it is said that the the set of real numbers is closed under the operation of. multiplication There is simply no way to escape the set of real numbers when multiplying. Mathematics Division,IMSP,UPLB 8

9 Time to think Consider the set E of even numbers. 1. Is E closed under addition? 2. Is E closed under multiplication? 2m+2n=2(m+n) (2m)(2n)=2(2mn) Consider the set O of odd numbers. 1. Is O closed under addition? 2. Is O closed under multiplication? 3+3=6 (2m+1)(2n+1)=? Mathematics Division,IMSP,UPLB 9

10 Time to think Consider the set Q of rational numbers. 1. Is Q closed under addition? 2. Is Q closed under multiplication? a b c d ad cb bd integer integer a b c d ac bd integer integer Mathematics Division,IMSP,UPLB 10

11 Time to think Consider the set Q c of irrational numbers. 1. Is Q c closed under addition? 2. Is Q c closed under multiplication? Mathematics Division,IMSP,UPLB 11

12 Time to think Is P (set of prime numbers) closed under addition? multiplication? Consider 2+2 and 2(2) Is C (set of composite numbers) closed under multiplication? OBVIOUS! RECALL Subtraction and Division: Is N closed under subtraction? Is Z closed under division? Mathematics Division,IMSP,UPLB 12

13 Note: We can define subtraction in terms of addition. Example: ( 2) ( 3) 1 BUT WE NEED TO SATISFY FIRST AN IMPORTANT PROPERTY: EXISTENCE OF ADDITIVE INVERSE Mathematics Division,IMSP,UPLB 13

14 Note: We can define division in terms of multiplication. But remember, we cannot divide by zero! Examples: BUT WE NEED TO SATISFY FIRST AN IMPORTANT PROPERTY: EXISTENCE OF MULTIPLICATIVE INVERSE Mathematics Division,IMSP,UPLB 14

15 Binary Operation Motivation: We can define our own operation. A (closed) binary operation * is a type of operation where there are two operands such that 1) The operands come from the same (non-empty) set 2) The set where the operands came from is closed under the operator 3) The result of the operation is unique x * y = z Operands Operator Mathematics Division,IMSP,UPLB Result 15

16 Binary Operation Examples + 2 is a binary operation on Z 2 ={0,1} Z 2 x Z 2 Z 2 (0,0) (1,0) (0,1) (1,1) = = = = 0 Mathematics Division,IMSP,UPLB 16

17 Binary Operation Examples Followed by is a binary operation on DC. The usual + and x are binary operations on R. Subtraction is not a binary operation on N but a binary relation on? Division is not a binary operation on R but a binary operation on R {0}. Exponentiation is a binary operation on N but not on R, why?

18 Binary Operation on R The convention is to operate from left to right =(3+2) = (5+6)+0+1 = (11+0)+1 = 11+1 = 12. We usually follow the order of operation: PEMA (parentheses, exponents, multiplications, additions).

19 Properties of Real Numbers Mathematics Division,IMSP,UPLB 19

20 In this lesson we look at some properties that apply to all real numbers. If you learn these properties, they will help you solve problems in algebra. Let's look at each property in detail, and apply it to an algebraic expression. Mathematics Division,IMSP,UPLB 20

21 Closure (Addition and Multiplication) Associativity (Addition and Multiplication) Existence of Identity (Addition and Multiplication) Existence of Inverses (Addition and Multiplication) Why this set of axioms are called Field Axioms? Commutativity (Addition and Multiplication) Distributivity (Multiplication over Addition) Mathematics Division,IMSP,UPLB 21

22 1. Commutative property a) Addition: For all real numbers a, b a + b = b + a -we can add numbers in any order b) Multiplication: For all real numbers a, b a x b = b x a -we can multiply numbers in any order Mathematics Division,IMSP,UPLB 22

23 Time to think a.) Is subtraction commutative in R? b.) Is division commutative in R? Mathematics Division,IMSP,UPLB 23

24 2. Associative property a) Addition: For all real numbers a, b, c, a + (b + c) = (a + b) + c we can cluster numbers in a sum in any way we want and still get the same answer b) Multiplication: For all real numbers a, b, c (a x b) x c = a x (b x c) we can cluster numbers in a product in any way we want and still get the same answer Mathematics Division,IMSP,UPLB 24

25 Time to think a.) Is subtraction associative in R? b.) Is division associative in R? Mathematics Division,IMSP,UPLB 25

26 3. Distributive property of multiplication over addition For all real numbers a, b, c a(b + c) = ab + ac (left-hand distributive law) and (a + b)c = ac + bc (right-hand distributive law) Mathematics Division,IMSP,UPLB 26

27 4. Existence of Unique Identity Element a) Addition: There exists a real number 0 such that for every real a, a + 0 = 0 + a = a - Zero added to any number is the number itself. - 0 is called the additive identity b) Multiplication: There exists a real number 1 such that for every real a, a x 1 = 1 x a = a - Any number multiplied by 1 gives the number itself. - 1 is called the multiplicative identity Mathematics Division,IMSP,UPLB 27

28 It is important to first have an identity element before having inverses 5. Existence of Inverses a) Additive Inverse (Opposite Sign) For every real number a there exists a real number, denoted (-a), such that a + (-a) = (-a) + a = 0 b) Multiplicative Inverse (Reciprocal) For every real number a except 0 there exists a real number, denoted by 1/ a, such that a x (1/ a) = (1/a) x a = 1 Mathematics Division,IMSP,UPLB 28

29 Examples 1. What is the additive inverse of 0? by Existence of 00 Additive Identity 2. What is the additive inverse of a?

30 (not part of the field axioms) We can always find another real number that lies between any two real numbers. Mathematics Division,IMSP,UPLB 30

31 SUMMARY Real numbers satisfy the field axioms. We need to be familiar with the properties of R in order to solve algebra problems. Do you think we can solve algebra problems without these properties? Mathematics Division,IMSP,UPLB 31

32 Exercises 1.Name all subsets of R to which these numbers belong: a. 12 d g. /3 b. 23 e. 27/3 h. e c. 0 f Estimate the position of the numbers in #1 on the number line. Mathematics Division,IMSP,UPLB 32

33 Exercises 3. Let U=R a) W Z d) Z + N g) W - N b) Z R e) W c Z h) R c c) N Q f) Z N i) Q c - Q Mathematics Division,IMSP,UPLB 33

34 Exercises 4) State the property of R that justifies the truth of the following statements: a) 3 f) (e + ) + 2 = e + ( + 2) 7 3R b) g) There are infinitely many R real numbers between c) = and 2.1 d) (2+1)+5=5+(2+1) h) 23 + (-23) = 0 e) 2(3+2)=6+4 i) 7(1/7) = 1 Answer is Commutativity not Associativity! Mathematics Division,IMSP,UPLB 34

35 Reflections 1. What is a number? How is number useful to our daily lives? 2. Name the subsets of the set of real numbers. 3. What are the nice properties that the set of real numbers obey under the operations of addition and multiplication? 4. Why do we consider these properties as nice? Mathematics Division,IMSP,UPLB 35

36 Equality Axioms Mathematics Division,IMSP,UPLB 36

37 Equality Axioms Reflexive Property of Equality For any a R, a a. Symmetric Property of Equality For a, b R, if a b then b a. Transitive Property of Equality For a, b, c R, if a b and b c then a c.

38 Equality Axioms Addition Property of Equality APE For a, b, c R, if a b then a c b c Multiplication Property of Equality MPE For a, b, c R, if a b then a c bc

39 Substitution If two real numbers are equal, then one may be substituted for the other in any algebraic expression. If x 5 y, then 2x 3y 25y 3 y. 3 3 Also, x 5. y

40 Solvable equations and Groups Mathematics Division,IMSP,UPLB 40

41 Solvable Equation Consider 3 x 5. x 2 is a solution and 2N. The equation is solvable in N. Now consider 3 x 0. This equation is not solvable in N.

42 Remark: N is not large enough to contain the solutions even for such simple linear equations that we have seen. Something must be done!

43 Solvable Equation Is 3 x 3 solvable in N? in W? Is 3 x 2 solvable in N? in W? in Z? FYI: [Z,+] is a group. But what is a group?

44 What properties of reals are needed to solve any linear equation of the form a+x=b? - Closure (under +) - Associativity (for +) - Existence of additive identity - Existence of additive inverse

45 a+x = b (a+x)+(-a) = b+(-a) APE (a+(-a))+x = b+(-a) Associativity (+) 0+x = b+(-a) Exist. of Add. Inverse x = b+(-a) Exist. of Add. Identity

46 Group (an algebraic structure) Given a non-empty set on is a G, the mathematical system gro up if 1. G is closed under. 2. is associative in G. G and an operation G, * is a binary operation on G 3. There is an identity element under in G. 4. Every element of G has an inverse under.

47 Examples Which of the following mathematical systems are groups? 1. N, Is N closed under +? Is + associative in N? What is the identity element? Therefore, N, is not a group.

48 Examples 2. Z, Is Z closed under? Is associative in Z? Counterexample Therefore, Z, is not a group.

49 Examples 3. N, 4. Q, 5. Z, 6. [Q { 0}, ]

50 TO DO Determine the identity element and inverse of each element of [DC, followed by ].

51 Any linear equation of the form a*x=b is solvable if a,bϵg and [G,*] is a group.

52 Abelian Group A group is an abelian group if its operation is commutative.

53 Example Z, is a group. Is commutative on Z? Therefore, Z, is an abelian group.

54 Solvable Equation Is 3x 6 solvable in Z? Is 3x 1 solvable in Z? We need multiplicative inverses!

55 Solvable Equations Is 2x 3 0 solvable in Z? in Q? in R? We need multiplicative inverses and we have TWO operations! Remark: ax+b=0 is solvable in a field. But before we discuss the concept of a field, we will discuss first the concept of a ring. Rings and Fields are also algebraic structures.

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