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Numbers & Sets, Functions, Operations with Functions, Decomposition of Functions

   

Added on  2020-11-09

19 Pages6630 Words153 Views
Lecture 1:Numbers & SetsA set is a collection of things (usually numbers).Definition (Formalnotation)A = {x : x satisfies some condition(s)}ExampleA = {x : x is an even number}B = {x : x is realand 1 < x < 2}Sometimes we can list the elements.ExampleC = {2, 3, 5, 7, 11, 13}a finite setD = {2, 3, 5, 7, 11, 13, ...}the infinite set of primesL1: 1 / 8Two important symbols are2is an element of⇢ is a subset of (is a part of)“2” expresses the relationship between an element and a set.“⇢” expresses the relationship between two sets.As a comparison, considerC ⇢ D1 < 2C ✓ D1 2ExerciseWrite in the appropriate symbol(a) 2A(b) CD(c)⇡2B(d) {2, 4, 6}AL1: 2 / 8Two set operations are\Intersection (where sets overlap)[Union (combination of sets)ExerciseEvaluate the operations(a) A [ C =(b) C \ D =(c) B \ A =(d) B [ D =L1: 3 / 8Standard SetsN-The set of NaturalNumbersZ -The set of IntegersQ -The set of RationalNumbersQ =npq: p, q 2 Z and q 6= 0oP -The set of IrrationalNumbersFor examplee = 2.7182818459...,p2 = 1.41421356..., ⇡ = 3.14159265...R -The set of RealNumbersWe have N ✓ Z ✓ Q ✓ R and Q [ P = RL1: 4 / 8

Some other notation we adopt:Z+-The set of non-negative IntegersZ+= {0, 1, 2, 3, 4, ...} = {0} [ NR+-The set of non-negative RealsR+= {x 2 R : x0}We can use intervalnotation for continuous sets of realnumbers.Example1.{x 2 R : 1 x < 5} = [1, 5)2.{x 2 R : x < 2} = (1, 2)3.R+= [0, 1)Note:Square brackets include the end value, round bracketsexclude the end value.You cannot put a square bracket around 1or1 as they are not numbers.L1: 5 / 8Absolute ValueThe absolute value of a realnumber is written as|x | =(x ,if x0x ,if x < 0and represents the distance of x from zero.Example|22| = 22|3| = 3Graphically, y = |x | is given byL1: 6 / 8DiscussionTrue of False?1.px2= x2.(px )2= xIntervals with absolute valuesThe set {x 2 R : |xa| b} represents the interval[ab, a + b].Example1.A = {x 2 R : |x| 1} =2.B = {x 2 R : |x1| < 3} =L1: 7 / 8The set {x 2 R : |xa| > b} represents allreals except theinterval[ab, a + b].ExampleC = {x 2 R : |x + 1| > 2} =L1: 8 / 8

Lecture 2:Functions (Section 2)DefinitionA function f is a rule or procedure which, for each number x insome subset A of R, assigns one and only one number f (x) in R.L2: 1 / 10ExerciseLet f (x ) =px24(a) Evaluatef (10)f (3)f (t2)f (x + 2)(b) Are there any values of x 2 R for which fis invalid?L2: 2 / 10Common functionsf (x) = cConstant functionf (x) = mx + b,m 6= 0Linear functionf (x) = xIdentity functionf (x) = a0+ a1x + a2x2+ ... + anxnPolynomialfunctionof degree nQuestion Does x2+ y2= a2represent a common function?L2: 3 / 10Notationf: A ! Rreads as “fis a mapping (function) from set A into set R”.The set A is called the domain of f, denoted Dom f .Informally,the domain represents values that are put into a function.NotationIf f: A ! R is a function, we define the range of f, denoted byRange f , byRange f= {y 2 R : y = f (x) for some x 2 A}Informally, the range represents the output values of the function.Finding the range is sometimes difficult.Inspection is one method,solving for x is another.Sketching a graph can be helpful.L2: 4 / 10

Example1.Let g : R ! R be given by g (x) = x2.Dom g=Range g=2.Let f1: Z ! R be given by f1(n) = 2n1.Dom f1=Range f1=L2: 5 / 10There are two parts to a function, the domain and the rule.Changing the rule or the domain changes the function.ExampleLet f2: R ! R be given by f2(x) = 2x1.Dom f2=Range f2=This function is di↵erent to f1in the previous example, as thedomains are di↵erent.DefinitionWe say that two functions fand g are equalif and only ifDom f= Dom g and f (x) = g (x) for each x 2 Dom f .L2: 6 / 10If the domain of a function is not given, it is assumed to be thelargest possible subset of R for which the given rule makes sense.This is often called the naturaldomain.Notes:denominators cannot be 0can only take square roots of non-negative numberscan only take logarithms of positive numbersExampleEarlier we considered f (x ) =px24.x240=) x24=) |x|2Dom f={x 2 R : |x |2}=(1,2] [ [2, 1).Note that Range f= R+.L2: 7 / 10ExerciseFind the domain and range for each of the following functions.(a) f1(x) =1x1(b) f2(x) = x + 2(c) f3(x) = 1 +px + 1(d) f4(x) =x24x2Are f2and f4equalfunctions?L2: 8 / 10

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