Mathematics In The Modern World PDF

MATHEMATICS IN THE MODERN WORLD MATHEMATICAL LANGUAGE AND SYMBOLS A R J A Y T. A LT O VA R M AT H I N S T R U C T O R MAIN IDEAS M AT H E M AT IC AL L A N GU AGE AN D S Y M BOL S M AT H E M AT IC AL E XP R E S S ION A N D S E N T E N C E S C ON V E N TION S IN M AT H E M AT...

MATHEMATICS IN THE MODERN WORLD MATHEMATICAL LANGUAGE AND SYMBOLS A R J A Y T. A LT O VA R M AT H I N S T R U C T O R MAIN IDEAS M AT H E M AT IC AL L A N GU AGE AN D S Y M BOL S M AT H E M AT IC AL E XP R E S S ION A N D S E N T E N C E S C ON V E N TION S IN M AT H E M AT IC A L L AN G U AGE F OU R B AS IC C ON C E P T S ON M AT H E M AT IC AL L AN GU AGE : (S E T S , FU N C T ION S , R E L AT I ON S , B IN ARY OP E RAT I ON S ) MATHEMATICAL LANGUAGE IS T H E SY S T E M U S E D T O C O M MU N IC AT E M AT H E M AT IC A L ID E A S  EXAMPLES OF MATHEMATICS LANGUAGE Mathematical Digits Greek Alphabet Symbols  CHARACTERISTICS OF MATHEMATICAL LANGUAGE 1. Mathematical Language is non-temporal (it has no past, present and future) 2. Mathematical Language carries no emotional content (it has no equivalent words for joy or sadness) 3. Mathematical Language is precise (statements are exact and accurate) 4. Mathematical Language is concise (no unnecessary words) 5. Mathematical Language is powerful  MATHEMATICAL EXPRESSION VS. MATHEMATICAL SENTENCES Mathematical Mathematical Sentences Expression -a statement about two -a finite combination of expressions, either using symbols that is well-defined numbers, variables, or a according to rules that combination of both. depend on the context -states a complete thought -does not contain a complete thought -can be determined if it is true -cannot be determined if it is or false true or false Example: 3+4=7, 3x+5=9, MATHEMATICAL CONVENTION A FAC T, N AM E , N O TAT I O N , O R U SAG E W H IC H IS G E N E R A LLY AG R E E D U P O N BY M AT H E M AT IC IA N S.  EXAMPLES OF MATHEMATICS CONVENTION PEMDA Formulas written from left to right S Mathematical Greek Letters FOUR BASIC CONCEPTS IN MATHEMATICAL LANGUAGE 1. SETS 2. FUNCTIONS 3. RELATIONS 4. BINARY OPERATIONS SETS A set is a well-defined collection of objects. The objects are called the elements or members of the set. Roster Method Rule Method The elements of the set A descriptive phrase is are enumerated and used to describe the separated by a comma elements or members of it is also called the set it is also called set tabulation method. builder notation, symbol it is written as {x P(x)}.  EXAMPLES OF SETS ROSTER METHOD RULE METHOD A = {1,2,3,5,8,13,21,…} A = {xx is a positive integer less than 10} B={ B = {xxis a letter in the word C = {d, i, r, t} dirt} D = {a, e, i, o, u} C = {xx is an integer, 1  x  8} D = {x}  EXERCISE WRITE THE FOLLOWING SET WRITE THE FOLLOWING SET USING USING RULE METHOD ROSTER METHOD A = {S,M,I,L,E} A = {xx is a letter in the word mathematics} B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, B = {xx is a positive integer, 3  x  Saturday 8} C = {1, 4, 9, 16, 25, 36, 49, 64, C = {x x = 2n + 3, n is a positive 81, 100} integer}  EXERCISE WRITE THE FOLLOWING SET WRITE THE FOLLOWING SET USING USING RULE METHOD ROSTER METHOD A = {S,M,I,L,E} A = {xx is a letter in the word A = {xx is the set of letters in the word mathematics} “Smile”} A = {m, a, t, h, e, i, c, s.} B = {Sunday, Monday, Tuesday, B = {xx is a positive integer, 3  x  Wednesday, Thursday, Friday, 8} Saturday B = {3, 4, 5, 6, 7, 8} B = {xx is the set of days in a week} C = {x x = 2n + 3, n is a positive C = {1, 4, 9, 16, 25, 36, 49, 64, integer} 81, 100,…} C = {5, 7, 9, 11, 13, …} C = {xx is the set of perfect squares}  SOME TERMS ON SETS Finite set is a set whose elements are limited or countable, and the last element can be identified. Example: A = {xx is a positive integer less than 10} C = {d, i, r, t} Infinite set is a set whose elements are unlimited or uncountable, and the last element cannot be specified. Example: F = {…, –2, –1, 0, 1, 2,…} G = {xx is a set of whole numbers}  SOME TERMS ON SETS A unit set is a set with only one element it is also called singleton. Example: I = {xx is a whole number greater than 1 but less than 3} J = {w} An empty set is a unique set with no elements (or null set), it is denoted by the symbol  or { }. L={xx is an integer less than 2 but greater than 1} M={xx is a number of panda bear in Manila Zoo}  SOME TERMS ON SETS Universal set is all sets under investigation in any application of set theory are assumed to be contained in some large fixed set, denoted by the symbol U. Example: U = {xx is a positive integer, x2 = 4} U = {1, 2, 3,…,100} U = {xx is an animal in Manila Zoo}  SOME TERMS ON SETS The cardinal number of a set is the number of elements or members in the set, the cardinality of set A is denoted by n(A) Example: Determine its cardinality of the Answe ff. sets r a. E = {a, e, i, o, u}, n(E) = 5 b. A = {xx is a positive integer less than n(A) 10} = 9 A = {1, 2, 3, 4, 5, 6, 7, c. C = {d, i, r, t} n(C) = 4 8, 9}  OPERATIONS ON SETS UNION INTERSECTION COMPLEMENT DIFFERENCE SYMMETRIC DIFFERENCE *DISJOINT SETS  UNION The union of A and B, denoted AB, is the set of all elements x in U such that x is in A or x is in B. Symbolically: AB = {xx  A  x  B}. INTERSECTI ON The intersection of A and B, denoted AB, is the set of all elements x in U such that x is in A and x is in B. Symbolically: AB = {xx  A  x  B}. COMPLEMEN T The complement of A (or absolute complement of A), denoted A’, is the set of all elements x in U such that x is not in A. Symbolically: A’ = {x  U  x  A}. DIFFERENCE The difference of A and B (or relative complement of B with respect to A), denoted A  B, is the set of all elements x in U such that x is in A and x is not in B. Symbolically: A  B = {xx  A  x  B} = AB’.  SYMMETRIC DIFFERENCE If set A and B are two sets, their symmetric difference as the set consisting of all elements that belong to A or to B, but not to both A and B. Symbolically: A  B = {xx  (AB)  x(AB)} = (AB)(AB)’ or (AB)  (AB). DISJOINT SETS Two sets are called disjoint (or non-intersecting) if and only if, they have no elements in common. Symbolically: A and B are disjoint  AB = .  EXAMPLE Suppose A = {a, b, c} B = {c, d,U = {a, b, c, d, e, f, g} e} Find the following a. (AB) B b. (AB)~U c. (A’ ~A)  B d. A  (B U) e. (A  B) (U~A’)  EXERCISE (20 minutes) If A = {a, b, c, d, e}, B = {a, e, i, o, u}, U = {a, b, c, d, e, f, g, h, i, j, k, l, o, u}. Perform the following operations on sets and find the solutions. a) A ∪ B b) A ∩ B c) A′ d) A - B e) U-(A-B)′ f) (A U B)′ U (A ∩ B)  KINDS OF SETS SUBSET PROPER SUBSET EQUAL SET POWER SET CARTESIAN PRODUCT SUBSET If A and B are sets, A is called subset of B, if and only if, every element of A is also an element of B. Symbolically: A  B  x, x  A x  B. Example: Suppose A = {c, d, e} B = {a, b, c, d, e} U = {a, b, c, d, e, f, g} Then A  B, since all elements of A is in B. PROPER SUBSET Let A and B be sets. A is a proper subset of B, if and only if, every element of A is in B but there is at least one element of B that is not in A. The symbol  denotes that it is not a proper subset. Symbolically: A  B  x, x  A x  B. Example: Suppose A = {c, d, e}= {a, b, c, d, B e} C = {e, a, c, b, d} U = {a, b, c, d, e, f, g} Then A  B, since all elements of A is in B. EQUAL SETS Given set A and B, A equals B, written, if and only if, every element of A is in B and every element of B is in A. Symbolically: A = B  A  B  B  A. Example: Suppose A = {a, b, c, d, e}, B = {a, b, d, e, c} U = {a, b, c, d, e, f, g} Then then A  B and B  A, thus A = B. EQUAL SETS Given a set S from universe U, the power set of S denoted by (S), is the collection (or sets) of all subsets of S. Example: Determine the power set of (a) A = {e, f}, (b) B = {1, 2, 3}. (a) A = (A) = {{e}, {f}, {e, {e, f} f}, } (b) B = {1, (B) = {{1}, {2}, {3}, {1, 2}, {1, 3}, 2, 3} {2, 3}, {1, 2, 3}, }. CARTESIAN PRODUCT The Cartesian product of sets A and B, written AxB, is AxB = {(a, b)  a  A and b  B} Exampl Let A = {2, 3, 5} and B = {7, 8}. Find each set. e: a. = {(2, 7), (2, 8), (3, 7), (3, 8), (5, 7), (5, 8)} AxB b. = BxA {(7, 2), (7, 3), (7, 5), (8, 2), (8, 3), (8, 5)} c. AxA = {(2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 3), (5, 5)} FUNCTIONS, RELATIONS AND BINARY OPERATIONS RELATIONS A collection of ordered pairs is called a relation. Example: The collection of ordered pairs R = { (0,1), (0, 2), (3, 4) } is a relation "0 is related to 1“ "0 is also related to 2“ "3 is related to 4" DOMAIN AND RANGE Let R be a relation from set A to the set B. domain of R is the set dom R dom R = {a  A (a, b)  R for some b  B}. image (or range) of R im R = {b  B (a, b)  R for some a  A}. Example: Find the domain and range of R = { (0,1), (0, 2), (3, 4) } Domain = { 0, 3 } Range = {1, 2, 4 } FUNCTIONS A function is a relation where each input has exactly one output. Example: Determine whether each of the following relations is a function. A = {(1, 3), (2, 4), (3, 5), (4, 6)} B = {(–2, 7), (–1, 3), (0, 1), (1, 5), (2, 5)} C = {(3, 0), (3, 2), (7, 4), (9, 1)} BINARY OPERATIONS Let S be a non-empty set, and ⋆ said to be a binary operation on S, if a ⋆ b is defined for all a, b ∈ S. In other words, ⋆ is a rule for any two elements in the set S. Binary operations require at least two inputs as it is defined from the cartesian product of set to set itself. Some fundamental binary operations are addition, subtraction, multiplication, and division. The inputs are known as the operands. Binary operations also have several properties like closure property, associative property, commutative property, identity element, and inverse element. ORDER OF OPERATIONS: PEMDAS P – Parentheses [{()}] E – Exponents (Powers and Roots) MD- Multiplication and Division (left to right) (× and ÷) AS – Addition and Subtraction (left to right) (+ and -) Example: 1. 60 ÷ (4 x 5) + 32 2. 12÷(1+3)−9×6 3. 9×8+4−2÷(4−2) 4. (51+78)÷32×(2−20) 5. [25 + {14 – (3 x 6)}] END OF PRESENTATION