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Functions

This document provides an introduction to functions and their properties. It defines what a function is as a mapping from a domain set to a codomain set, and introduces related concepts like domain, codomain, range, and the notation for functions. It then gives examples of specifying functions explicitly and with formulas. The document discusses properties of functions like injectivity, surjectivity, and being increasing or decreasing. It provides examples of determining if a function has these properties. The document concludes by introducing the concept of the inverse function for bijective functions.

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Fall 2002 CMSC 203 - Discrete Structures 2 …… and the following mathematicaland the following mathematical appetizer is about…appetizer is about… FunctionsFunctions
Fall 2002 CMSC 203 - Discrete Structures 3 FunctionsFunctions AA functionfunction f from a set A to a set B is anf from a set A to a set B is an assignmentassignment of exactly one element of B to eachof exactly one element of B to each element of A.element of A. We writeWe write f(a) = bf(a) = b if b is the unique element of B assigned by theif b is the unique element of B assigned by the function f to the element a of A.function f to the element a of A. If f is a function from A to B, we writeIf f is a function from A to B, we write f: Af: A→→BB (note: Here, “(note: Here, “→→“ has nothing to do with if… then)“ has nothing to do with if… then)
Fall 2002 CMSC 203 - Discrete Structures 4 FunctionsFunctions If f:AIf f:A→→B, we say that A is theB, we say that A is the domaindomain of f and Bof f and B is theis the codomaincodomain of f.of f. If f(a) = b, we say that b is theIf f(a) = b, we say that b is the imageimage of a and a isof a and a is thethe pre-imagepre-image of b.of b. TheThe rangerange of f:Aof f:A→→B is the set of all images ofB is the set of all images of elements of A.elements of A. We say that f:AWe say that f:A→→BB mapsmaps A to B.A to B.
Fall 2002 CMSC 203 - Discrete Structures 5 FunctionsFunctions Let us take a look at the function f:PLet us take a look at the function f:P→→C withC with P = {Linda, Max, Kathy, Peter}P = {Linda, Max, Kathy, Peter} C = {Boston, New York, Hong Kong, Moscow}C = {Boston, New York, Hong Kong, Moscow} f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = New Yorkf(Peter) = New York Here, the range of f is C.Here, the range of f is C.
Fall 2002 CMSC 203 - Discrete Structures 6 FunctionsFunctions Let us re-specify f as follows:Let us re-specify f as follows: f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Bostonf(Peter) = Boston Is f still a function?Is f still a function? yesyes {Moscow, Boston, Hong Kong}{Moscow, Boston, Hong Kong}What is its range?What is its range?
Fall 2002 CMSC 203 - Discrete Structures 7 FunctionsFunctions Other ways to represent f:Other ways to represent f: BostonBostonPeterPeter HongHong KongKong KathyKathy BostonBostonMaxMax MoscowMoscowLindaLinda f(x)f(x)xx LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow
Fall 2002 CMSC 203 - Discrete Structures 8 FunctionsFunctions If the domain of our function f is large, it isIf the domain of our function f is large, it is convenient to specify f with aconvenient to specify f with a formulaformula, e.g.:, e.g.: f:f:RR→→RR f(x) = 2xf(x) = 2x This leads to:This leads to: f(1) = 2f(1) = 2 f(3) = 6f(3) = 6 f(-3) = -6f(-3) = -6 ……
Fall 2002 CMSC 203 - Discrete Structures 9 FunctionsFunctions Let fLet f11 and fand f22 be functions from A tobe functions from A to RR.. Then theThen the sumsum and theand the productproduct of fof f11 and fand f22 are alsoare also functions from A tofunctions from A to RR defined by:defined by: (f(f11 + f+ f22)(x) = f)(x) = f11(x) + f(x) + f22(x)(x) (f(f11ff22)(x) = f)(x) = f11(x) f(x) f22(x)(x) Example:Example: ff11(x) = 3x, f(x) = 3x, f22(x) = x + 5(x) = x + 5 (f(f11 + f+ f22)(x) = f)(x) = f11(x) + f(x) + f22(x) = 3x + x + 5 = 4x + 5(x) = 3x + x + 5 = 4x + 5 (f(f11ff22)(x) = f)(x) = f11(x) f(x) f22(x) = 3x (x + 5) = 3x(x) = 3x (x + 5) = 3x22 + 15x+ 15x
Fall 2002 CMSC 203 - Discrete Structures 10 FunctionsFunctions We already know that theWe already know that the rangerange of a functionof a function f:Af:A→→B is the set of all images of elements aB is the set of all images of elements a∈∈A.A. If we only regard aIf we only regard a subsetsubset SS⊆⊆A, the set of allA, the set of all images of elements simages of elements s∈∈S is called theS is called the imageimage of S.of S. We denote the image of S by f(S):We denote the image of S by f(S): f(S) = {f(s) | sf(S) = {f(s) | s∈∈S}S}
Fall 2002 CMSC 203 - Discrete Structures 11 FunctionsFunctions Let us look at the following well-known function:Let us look at the following well-known function: f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Bostonf(Peter) = Boston What is the image of S = {Linda, Max} ?What is the image of S = {Linda, Max} ? f(S) = {Moscow, Boston}f(S) = {Moscow, Boston} What is the image of S = {Max, Peter} ?What is the image of S = {Max, Peter} ? f(S) = {Boston}f(S) = {Boston}
Fall 2002 CMSC 203 - Discrete Structures 12 Properties of FunctionsProperties of Functions A function f:AA function f:A→→B is said to beB is said to be one-to-oneone-to-one (or(or injectiveinjective), if and only if), if and only if ∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y) In other words:In other words: f is one-to-one if and only if itf is one-to-one if and only if it does not map two distinct elements of A onto thedoes not map two distinct elements of A onto the same element of B.same element of B.
Fall 2002 CMSC 203 - Discrete Structures 13 Properties of FunctionsProperties of Functions And again…And again… f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Bostonf(Peter) = Boston Is f one-to-one?Is f one-to-one? No, Max and Peter areNo, Max and Peter are mapped onto the samemapped onto the same element of the image.element of the image. g(Linda) = Moscowg(Linda) = Moscow g(Max) = Bostong(Max) = Boston g(Kathy) = Hong Kongg(Kathy) = Hong Kong g(Peter) = New Yorkg(Peter) = New York Is g one-to-one?Is g one-to-one? Yes, each element isYes, each element is assigned a uniqueassigned a unique element of the image.element of the image.
Fall 2002 CMSC 203 - Discrete Structures 14 Properties of FunctionsProperties of Functions How can we prove that a function f is one-to-one?How can we prove that a function f is one-to-one? Whenever you want to prove something, first takeWhenever you want to prove something, first take a look at the relevant definition(s):a look at the relevant definition(s): ∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y) Example:Example: f:f:RR→→RR f(x) = xf(x) = x22 Disproof by counterexample: f(3) = f(-3), but 3f(3) = f(-3), but 3 ≠≠ -3, so f is not one-to-one.-3, so f is not one-to-one.
Fall 2002 CMSC 203 - Discrete Structures 15 Properties of FunctionsProperties of Functions …… and yet another example:and yet another example: f:f:RR→→RR f(x) = 3xf(x) = 3x One-to-one:One-to-one: ∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y) To show:To show: f(x)f(x) ≠≠ f(y) whenever xf(y) whenever x ≠≠ yy xx ≠≠ yy ⇔ 3x3x ≠≠ 3y3y ⇔ f(x)f(x) ≠≠ f(y),f(y), so if xso if x ≠≠ y, then f(x)y, then f(x) ≠≠ f(y), that is, f is one-to-one.f(y), that is, f is one-to-one.
Fall 2002 CMSC 203 - Discrete Structures 16 Properties of FunctionsProperties of Functions A function f:AA function f:A→→B with A,BB with A,B ⊆⊆ R is calledR is called strictlystrictly increasingincreasing, if, if ∀∀x,yx,y∈∈A (x < yA (x < y →→ f(x) < f(y)),f(x) < f(y)), andand strictly decreasingstrictly decreasing, if, if ∀∀x,yx,y∈∈A (x < yA (x < y →→ f(x) > f(y)).f(x) > f(y)). Obviously, a function that is either strictlyObviously, a function that is either strictly increasing or strictly decreasing isincreasing or strictly decreasing is one-to-oneone-to-one..
Fall 2002 CMSC 203 - Discrete Structures 17 Properties of FunctionsProperties of Functions A function f:AA function f:A→→B is calledB is called ontoonto, or, or surjectivesurjective, if, if and only if for every element band only if for every element b∈∈B there is anB there is an element aelement a∈∈A with f(a) = b.A with f(a) = b. In other words, f is onto if and only if itsIn other words, f is onto if and only if its rangerange isis itsits entire codomainentire codomain.. A function f: AA function f: A→→B is aB is a one-to-one correspondenceone-to-one correspondence,, or aor a bijectionbijection, if and only if it is both one-to-one, if and only if it is both one-to-one and onto.and onto. Obviously, if f is a bijection and A and B are finiteObviously, if f is a bijection and A and B are finite sets, then |A| = |B|.sets, then |A| = |B|.
Fall 2002 CMSC 203 - Discrete Structures 18 Properties of FunctionsProperties of Functions Examples:Examples: In the following examples, we use the arrowIn the following examples, we use the arrow representation to illustrate functions f:Arepresentation to illustrate functions f:A→→B.B. In each example, the complete sets A and B areIn each example, the complete sets A and B are shown.shown.
Fall 2002 CMSC 203 - Discrete Structures 19 Properties of FunctionsProperties of Functions Is f injective?Is f injective? No.No. Is f surjective?Is f surjective? No.No. Is f bijective?Is f bijective? No.No. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow
Fall 2002 CMSC 203 - Discrete Structures 20 Properties of FunctionsProperties of Functions Is f injective?Is f injective? No.No. Is f surjective?Is f surjective? Yes.Yes. Is f bijective?Is f bijective? No.No. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow PaulPaul
Fall 2002 CMSC 203 - Discrete Structures 21 Properties of FunctionsProperties of Functions Is f injective?Is f injective? Yes.Yes. Is f surjective?Is f surjective? No.No. Is f bijective?Is f bijective? No.No. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLßßbeckbeck
Fall 2002 CMSC 203 - Discrete Structures 22 Properties of FunctionsProperties of Functions Is f injective?Is f injective? No! f is not evenNo! f is not even a function!a function! LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLßßbeckbeck
Fall 2002 CMSC 203 - Discrete Structures 23 Properties of FunctionsProperties of Functions Is f injective?Is f injective? Yes.Yes. Is f surjective?Is f surjective? Yes.Yes. Is f bijective?Is f bijective? Yes.Yes. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLßßbeckbeckHelenaHelena
Fall 2002 CMSC 203 - Discrete Structures 24 InversionInversion An interesting property of bijections is thatAn interesting property of bijections is that they have anthey have an inverse functioninverse function.. TheThe inverse functioninverse function of the bijection f:Aof the bijection f:A→→BB is the function fis the function f-1-1 :B:B→→A withA with ff-1-1 (b) = a whenever f(a) = b.(b) = a whenever f(a) = b.
Fall 2002 CMSC 203 - Discrete Structures 25 InversionInversion Example:Example: f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Lf(Peter) = Lßßbeckbeck f(Helena) = New Yorkf(Helena) = New York Clearly, f is bijective.Clearly, f is bijective. The inverse functionThe inverse function ff-1-1 is given by:is given by: ff-1-1 (Moscow) = Linda(Moscow) = Linda ff-1-1 (Boston) = Max(Boston) = Max ff-1-1 (Hong Kong) = Kathy(Hong Kong) = Kathy ff-1-1 (L(Lßßbeck) = Peterbeck) = Peter ff-1-1 (New York) = Helena(New York) = Helena Inversion is onlyInversion is only possible for bijectionspossible for bijections (= invertible functions)(= invertible functions)
Fall 2002 CMSC 203 - Discrete Structures 26 InversionInversion LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLüübeckbeckHelenaHelena ff ff-1-1 ff-1-1 :C:C→→P is noP is no function, becausefunction, because it is not definedit is not defined for all elements offor all elements of C and assigns twoC and assigns two images to the pre-images to the pre- image New York.image New York.
Fall 2002 CMSC 203 - Discrete Structures 27 CompositionComposition TheThe compositioncomposition of two functions g:Aof two functions g:A→→B andB and f:Bf:B→→C, denoted by fC, denoted by f°°g, is defined byg, is defined by (f(f°°g)(a) = f(g(a))g)(a) = f(g(a)) This means thatThis means that • firstfirst, function g is applied to element a, function g is applied to element a∈∈A,A, mapping it onto an element of B,mapping it onto an element of B, • thenthen, function f is applied to this element of, function f is applied to this element of B, mapping it onto an element of C.B, mapping it onto an element of C. • ThereforeTherefore, the composite function maps, the composite function maps from A to C.from A to C.
Fall 2002 CMSC 203 - Discrete Structures 28 CompositionComposition Example:Example: f(x) = 7x – 4, g(x) = 3x,f(x) = 7x – 4, g(x) = 3x, f:f:RR→→RR, g:, g:RR→→RR (f(f°°g)(5) = f(g(5)) = f(15) = 105 – 4 = 101g)(5) = f(g(5)) = f(15) = 105 – 4 = 101 (f(f°°g)(x) = f(g(x)) = f(3x) = 21x - 4g)(x) = f(g(x)) = f(3x) = 21x - 4
Fall 2002 CMSC 203 - Discrete Structures 29 CompositionComposition Composition of a function and its inverse:Composition of a function and its inverse: (f(f-1-1 °°f)(x) = ff)(x) = f-1-1 (f(x)) = x(f(x)) = x The composition of a function and its inverseThe composition of a function and its inverse is theis the identity functionidentity function i(x) = x.i(x) = x.
Fall 2002 CMSC 203 - Discrete Structures 30 GraphsGraphs TheThe graphgraph of a functionof a function f:Af:A→→B is the set ofB is the set of ordered pairs {(a, b) | aordered pairs {(a, b) | a∈∈A and f(a) = b}.A and f(a) = b}. The graph is a subset of AThe graph is a subset of A××B that can be usedB that can be used to visualize f in a two-dimensional coordinateto visualize f in a two-dimensional coordinate system.system.
Fall 2002 CMSC 203 - Discrete Structures 31 Floor and Ceiling FunctionsFloor and Ceiling Functions TheThe floorfloor andand ceilingceiling functions map the realfunctions map the real numbers onto the integers (numbers onto the integers (RR→→ZZ).). TheThe floorfloor function assigns to rfunction assigns to r∈∈RR the largestthe largest zz∈∈ZZ with zwith z ≤≤ r, denoted byr, denoted by rr.. Examples:Examples: 2.32.3 = 2,= 2, 22 = 2,= 2, 0.50.5 = 0,= 0, -3.5-3.5 = -4= -4 TheThe ceilingceiling function assigns to rfunction assigns to r∈∈RR the smallestthe smallest zz∈∈ZZ with zwith z ≥≥ r, denoted byr, denoted by rr.. Examples:Examples: 2.32.3 = 3,= 3, 22 = 2,= 2, 0.50.5 = 1,= 1, -3.5-3.5 = -3= -3
32 Do you find this slides were useful?Do you find this slides were useful? One second of your life , can bring a smile in a girl lifeOne second of your life , can bring a smile in a girl life If Yes ,JoinIf Yes ,Join Dreams SchoolDreams School “Campaign for Female Education”“Campaign for Female Education” Help us in bringing a change in a girl life, becauseHelp us in bringing a change in a girl life, because ““When someone takes away your pens youWhen someone takes away your pens you realize quite how important education is”.realize quite how important education is”. Just Click on any advertisement on the page, yourJust Click on any advertisement on the page, your one click can make her smileone click can make her smile.. Eliminate InequalityEliminate Inequality ““Not WomenNot Women”” One second of your life , can bring a smile in her life!!One second of your life , can bring a smile in her life!! Do you find these slides were useful?Do you find these slides were useful? If Yes ,JoinIf Yes ,Join Dreams SchoolDreams School ““CampaignCampaign for Female Educationfor Female Education”” Help us in bringing a change in a girl life,Help us in bringing a change in a girl life, becausebecause ““When someone takes awayWhen someone takes away your pens you realize quite howyour pens you realize quite how important education isimportant education is”.”. Just Click on any advertisement on the page,Just Click on any advertisement on the page, your one click can make her smileyour one click can make her smile.. We our doing our part & u ?We our doing our part & u ? Eliminate InequalityEliminate Inequality ““Not WomenNot Women””

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In this document
Powered by AI

Overview of functions, domain, codomain, and image of functions are introduced.

Functions with specific values assigned to elements from predefined sets are illustrated.

Various ways to represent functions, including formulas and mappings, are discussed.

Functions can be added or multiplied, and the range of a function is explored.

Concept of the image of a subset under a function using examples to find images.

Properties such as injectivity, surjectivity, and their implications are explained.

Practical examples of functions assessing if they are injective, surjective, and bijective.

Definition and examples of inverse functions; conditions for a function to have an inverse.

Definition and examples of function composition and its relation to inverse functions.

Floor and ceiling functions that map real numbers to integers with examples.

An appeal for supporting female education through the Dreams School initiative.

Functions

  • 1.
  • 2.
    Fall 2002 CMSC203 - Discrete Structures 2 …… and the following mathematicaland the following mathematical appetizer is about…appetizer is about… FunctionsFunctions
  • 3.
    Fall 2002 CMSC203 - Discrete Structures 3 FunctionsFunctions AA functionfunction f from a set A to a set B is anf from a set A to a set B is an assignmentassignment of exactly one element of B to eachof exactly one element of B to each element of A.element of A. We writeWe write f(a) = bf(a) = b if b is the unique element of B assigned by theif b is the unique element of B assigned by the function f to the element a of A.function f to the element a of A. If f is a function from A to B, we writeIf f is a function from A to B, we write f: Af: A→→BB (note: Here, “(note: Here, “→→“ has nothing to do with if… then)“ has nothing to do with if… then)
  • 4.
    Fall 2002 CMSC203 - Discrete Structures 4 FunctionsFunctions If f:AIf f:A→→B, we say that A is theB, we say that A is the domaindomain of f and Bof f and B is theis the codomaincodomain of f.of f. If f(a) = b, we say that b is theIf f(a) = b, we say that b is the imageimage of a and a isof a and a is thethe pre-imagepre-image of b.of b. TheThe rangerange of f:Aof f:A→→B is the set of all images ofB is the set of all images of elements of A.elements of A. We say that f:AWe say that f:A→→BB mapsmaps A to B.A to B.
  • 5.
    Fall 2002 CMSC203 - Discrete Structures 5 FunctionsFunctions Let us take a look at the function f:PLet us take a look at the function f:P→→C withC with P = {Linda, Max, Kathy, Peter}P = {Linda, Max, Kathy, Peter} C = {Boston, New York, Hong Kong, Moscow}C = {Boston, New York, Hong Kong, Moscow} f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = New Yorkf(Peter) = New York Here, the range of f is C.Here, the range of f is C.
  • 6.
    Fall 2002 CMSC203 - Discrete Structures 6 FunctionsFunctions Let us re-specify f as follows:Let us re-specify f as follows: f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Bostonf(Peter) = Boston Is f still a function?Is f still a function? yesyes {Moscow, Boston, Hong Kong}{Moscow, Boston, Hong Kong}What is its range?What is its range?
  • 7.
    Fall 2002 CMSC203 - Discrete Structures 7 FunctionsFunctions Other ways to represent f:Other ways to represent f: BostonBostonPeterPeter HongHong KongKong KathyKathy BostonBostonMaxMax MoscowMoscowLindaLinda f(x)f(x)xx LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow
  • 8.
    Fall 2002 CMSC203 - Discrete Structures 8 FunctionsFunctions If the domain of our function f is large, it isIf the domain of our function f is large, it is convenient to specify f with aconvenient to specify f with a formulaformula, e.g.:, e.g.: f:f:RR→→RR f(x) = 2xf(x) = 2x This leads to:This leads to: f(1) = 2f(1) = 2 f(3) = 6f(3) = 6 f(-3) = -6f(-3) = -6 ……
  • 9.
    Fall 2002 CMSC203 - Discrete Structures 9 FunctionsFunctions Let fLet f11 and fand f22 be functions from A tobe functions from A to RR.. Then theThen the sumsum and theand the productproduct of fof f11 and fand f22 are alsoare also functions from A tofunctions from A to RR defined by:defined by: (f(f11 + f+ f22)(x) = f)(x) = f11(x) + f(x) + f22(x)(x) (f(f11ff22)(x) = f)(x) = f11(x) f(x) f22(x)(x) Example:Example: ff11(x) = 3x, f(x) = 3x, f22(x) = x + 5(x) = x + 5 (f(f11 + f+ f22)(x) = f)(x) = f11(x) + f(x) + f22(x) = 3x + x + 5 = 4x + 5(x) = 3x + x + 5 = 4x + 5 (f(f11ff22)(x) = f)(x) = f11(x) f(x) f22(x) = 3x (x + 5) = 3x(x) = 3x (x + 5) = 3x22 + 15x+ 15x
  • 10.
    Fall 2002 CMSC203 - Discrete Structures 10 FunctionsFunctions We already know that theWe already know that the rangerange of a functionof a function f:Af:A→→B is the set of all images of elements aB is the set of all images of elements a∈∈A.A. If we only regard aIf we only regard a subsetsubset SS⊆⊆A, the set of allA, the set of all images of elements simages of elements s∈∈S is called theS is called the imageimage of S.of S. We denote the image of S by f(S):We denote the image of S by f(S): f(S) = {f(s) | sf(S) = {f(s) | s∈∈S}S}
  • 11.
    Fall 2002 CMSC203 - Discrete Structures 11 FunctionsFunctions Let us look at the following well-known function:Let us look at the following well-known function: f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Bostonf(Peter) = Boston What is the image of S = {Linda, Max} ?What is the image of S = {Linda, Max} ? f(S) = {Moscow, Boston}f(S) = {Moscow, Boston} What is the image of S = {Max, Peter} ?What is the image of S = {Max, Peter} ? f(S) = {Boston}f(S) = {Boston}
  • 12.
    Fall 2002 CMSC203 - Discrete Structures 12 Properties of FunctionsProperties of Functions A function f:AA function f:A→→B is said to beB is said to be one-to-oneone-to-one (or(or injectiveinjective), if and only if), if and only if ∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y) In other words:In other words: f is one-to-one if and only if itf is one-to-one if and only if it does not map two distinct elements of A onto thedoes not map two distinct elements of A onto the same element of B.same element of B.
  • 13.
    Fall 2002 CMSC203 - Discrete Structures 13 Properties of FunctionsProperties of Functions And again…And again… f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Bostonf(Peter) = Boston Is f one-to-one?Is f one-to-one? No, Max and Peter areNo, Max and Peter are mapped onto the samemapped onto the same element of the image.element of the image. g(Linda) = Moscowg(Linda) = Moscow g(Max) = Bostong(Max) = Boston g(Kathy) = Hong Kongg(Kathy) = Hong Kong g(Peter) = New Yorkg(Peter) = New York Is g one-to-one?Is g one-to-one? Yes, each element isYes, each element is assigned a uniqueassigned a unique element of the image.element of the image.
  • 14.
    Fall 2002 CMSC203 - Discrete Structures 14 Properties of FunctionsProperties of Functions How can we prove that a function f is one-to-one?How can we prove that a function f is one-to-one? Whenever you want to prove something, first takeWhenever you want to prove something, first take a look at the relevant definition(s):a look at the relevant definition(s): ∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y) Example:Example: f:f:RR→→RR f(x) = xf(x) = x22 Disproof by counterexample: f(3) = f(-3), but 3f(3) = f(-3), but 3 ≠≠ -3, so f is not one-to-one.-3, so f is not one-to-one.
  • 15.
    Fall 2002 CMSC203 - Discrete Structures 15 Properties of FunctionsProperties of Functions …… and yet another example:and yet another example: f:f:RR→→RR f(x) = 3xf(x) = 3x One-to-one:One-to-one: ∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y) To show:To show: f(x)f(x) ≠≠ f(y) whenever xf(y) whenever x ≠≠ yy xx ≠≠ yy ⇔ 3x3x ≠≠ 3y3y ⇔ f(x)f(x) ≠≠ f(y),f(y), so if xso if x ≠≠ y, then f(x)y, then f(x) ≠≠ f(y), that is, f is one-to-one.f(y), that is, f is one-to-one.
  • 16.
    Fall 2002 CMSC203 - Discrete Structures 16 Properties of FunctionsProperties of Functions A function f:AA function f:A→→B with A,BB with A,B ⊆⊆ R is calledR is called strictlystrictly increasingincreasing, if, if ∀∀x,yx,y∈∈A (x < yA (x < y →→ f(x) < f(y)),f(x) < f(y)), andand strictly decreasingstrictly decreasing, if, if ∀∀x,yx,y∈∈A (x < yA (x < y →→ f(x) > f(y)).f(x) > f(y)). Obviously, a function that is either strictlyObviously, a function that is either strictly increasing or strictly decreasing isincreasing or strictly decreasing is one-to-oneone-to-one..
  • 17.
    Fall 2002 CMSC203 - Discrete Structures 17 Properties of FunctionsProperties of Functions A function f:AA function f:A→→B is calledB is called ontoonto, or, or surjectivesurjective, if, if and only if for every element band only if for every element b∈∈B there is anB there is an element aelement a∈∈A with f(a) = b.A with f(a) = b. In other words, f is onto if and only if itsIn other words, f is onto if and only if its rangerange isis itsits entire codomainentire codomain.. A function f: AA function f: A→→B is aB is a one-to-one correspondenceone-to-one correspondence,, or aor a bijectionbijection, if and only if it is both one-to-one, if and only if it is both one-to-one and onto.and onto. Obviously, if f is a bijection and A and B are finiteObviously, if f is a bijection and A and B are finite sets, then |A| = |B|.sets, then |A| = |B|.
  • 18.
    Fall 2002 CMSC203 - Discrete Structures 18 Properties of FunctionsProperties of Functions Examples:Examples: In the following examples, we use the arrowIn the following examples, we use the arrow representation to illustrate functions f:Arepresentation to illustrate functions f:A→→B.B. In each example, the complete sets A and B areIn each example, the complete sets A and B are shown.shown.
  • 19.
    Fall 2002 CMSC203 - Discrete Structures 19 Properties of FunctionsProperties of Functions Is f injective?Is f injective? No.No. Is f surjective?Is f surjective? No.No. Is f bijective?Is f bijective? No.No. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow
  • 20.
    Fall 2002 CMSC203 - Discrete Structures 20 Properties of FunctionsProperties of Functions Is f injective?Is f injective? No.No. Is f surjective?Is f surjective? Yes.Yes. Is f bijective?Is f bijective? No.No. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow PaulPaul
  • 21.
    Fall 2002 CMSC203 - Discrete Structures 21 Properties of FunctionsProperties of Functions Is f injective?Is f injective? Yes.Yes. Is f surjective?Is f surjective? No.No. Is f bijective?Is f bijective? No.No. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLßßbeckbeck
  • 22.
    Fall 2002 CMSC203 - Discrete Structures 22 Properties of FunctionsProperties of Functions Is f injective?Is f injective? No! f is not evenNo! f is not even a function!a function! LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLßßbeckbeck
  • 23.
    Fall 2002 CMSC203 - Discrete Structures 23 Properties of FunctionsProperties of Functions Is f injective?Is f injective? Yes.Yes. Is f surjective?Is f surjective? Yes.Yes. Is f bijective?Is f bijective? Yes.Yes. LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLßßbeckbeckHelenaHelena
  • 24.
    Fall 2002 CMSC203 - Discrete Structures 24 InversionInversion An interesting property of bijections is thatAn interesting property of bijections is that they have anthey have an inverse functioninverse function.. TheThe inverse functioninverse function of the bijection f:Aof the bijection f:A→→BB is the function fis the function f-1-1 :B:B→→A withA with ff-1-1 (b) = a whenever f(a) = b.(b) = a whenever f(a) = b.
  • 25.
    Fall 2002 CMSC203 - Discrete Structures 25 InversionInversion Example:Example: f(Linda) = Moscowf(Linda) = Moscow f(Max) = Bostonf(Max) = Boston f(Kathy) = Hong Kongf(Kathy) = Hong Kong f(Peter) = Lf(Peter) = Lßßbeckbeck f(Helena) = New Yorkf(Helena) = New York Clearly, f is bijective.Clearly, f is bijective. The inverse functionThe inverse function ff-1-1 is given by:is given by: ff-1-1 (Moscow) = Linda(Moscow) = Linda ff-1-1 (Boston) = Max(Boston) = Max ff-1-1 (Hong Kong) = Kathy(Hong Kong) = Kathy ff-1-1 (L(Lßßbeck) = Peterbeck) = Peter ff-1-1 (New York) = Helena(New York) = Helena Inversion is onlyInversion is only possible for bijectionspossible for bijections (= invertible functions)(= invertible functions)
  • 26.
    Fall 2002 CMSC203 - Discrete Structures 26 InversionInversion LindaLinda MaxMax KathyKathy PeterPeter BostonBoston New YorkNew York Hong KongHong Kong MoscowMoscow LLüübeckbeckHelenaHelena ff ff-1-1 ff-1-1 :C:C→→P is noP is no function, becausefunction, because it is not definedit is not defined for all elements offor all elements of C and assigns twoC and assigns two images to the pre-images to the pre- image New York.image New York.
  • 27.
    Fall 2002 CMSC203 - Discrete Structures 27 CompositionComposition TheThe compositioncomposition of two functions g:Aof two functions g:A→→B andB and f:Bf:B→→C, denoted by fC, denoted by f°°g, is defined byg, is defined by (f(f°°g)(a) = f(g(a))g)(a) = f(g(a)) This means thatThis means that • firstfirst, function g is applied to element a, function g is applied to element a∈∈A,A, mapping it onto an element of B,mapping it onto an element of B, • thenthen, function f is applied to this element of, function f is applied to this element of B, mapping it onto an element of C.B, mapping it onto an element of C. • ThereforeTherefore, the composite function maps, the composite function maps from A to C.from A to C.
  • 28.
    Fall 2002 CMSC203 - Discrete Structures 28 CompositionComposition Example:Example: f(x) = 7x – 4, g(x) = 3x,f(x) = 7x – 4, g(x) = 3x, f:f:RR→→RR, g:, g:RR→→RR (f(f°°g)(5) = f(g(5)) = f(15) = 105 – 4 = 101g)(5) = f(g(5)) = f(15) = 105 – 4 = 101 (f(f°°g)(x) = f(g(x)) = f(3x) = 21x - 4g)(x) = f(g(x)) = f(3x) = 21x - 4
  • 29.
    Fall 2002 CMSC203 - Discrete Structures 29 CompositionComposition Composition of a function and its inverse:Composition of a function and its inverse: (f(f-1-1 °°f)(x) = ff)(x) = f-1-1 (f(x)) = x(f(x)) = x The composition of a function and its inverseThe composition of a function and its inverse is theis the identity functionidentity function i(x) = x.i(x) = x.
  • 30.
    Fall 2002 CMSC203 - Discrete Structures 30 GraphsGraphs TheThe graphgraph of a functionof a function f:Af:A→→B is the set ofB is the set of ordered pairs {(a, b) | aordered pairs {(a, b) | a∈∈A and f(a) = b}.A and f(a) = b}. The graph is a subset of AThe graph is a subset of A××B that can be usedB that can be used to visualize f in a two-dimensional coordinateto visualize f in a two-dimensional coordinate system.system.
  • 31.
    Fall 2002 CMSC203 - Discrete Structures 31 Floor and Ceiling FunctionsFloor and Ceiling Functions TheThe floorfloor andand ceilingceiling functions map the realfunctions map the real numbers onto the integers (numbers onto the integers (RR→→ZZ).). TheThe floorfloor function assigns to rfunction assigns to r∈∈RR the largestthe largest zz∈∈ZZ with zwith z ≤≤ r, denoted byr, denoted by rr.. Examples:Examples: 2.32.3 = 2,= 2, 22 = 2,= 2, 0.50.5 = 0,= 0, -3.5-3.5 = -4= -4 TheThe ceilingceiling function assigns to rfunction assigns to r∈∈RR the smallestthe smallest zz∈∈ZZ with zwith z ≥≥ r, denoted byr, denoted by rr.. Examples:Examples: 2.32.3 = 3,= 3, 22 = 2,= 2, 0.50.5 = 1,= 1, -3.5-3.5 = -3= -3
  • 32.
    32 Do you findthis slides were useful?Do you find this slides were useful? One second of your life , can bring a smile in a girl lifeOne second of your life , can bring a smile in a girl life If Yes ,JoinIf Yes ,Join Dreams SchoolDreams School “Campaign for Female Education”“Campaign for Female Education” Help us in bringing a change in a girl life, becauseHelp us in bringing a change in a girl life, because ““When someone takes away your pens youWhen someone takes away your pens you realize quite how important education is”.realize quite how important education is”. Just Click on any advertisement on the page, yourJust Click on any advertisement on the page, your one click can make her smileone click can make her smile.. Eliminate InequalityEliminate Inequality ““Not WomenNot Women”” One second of your life , can bring a smile in her life!!One second of your life , can bring a smile in her life!! Do you find these slides were useful?Do you find these slides were useful? If Yes ,JoinIf Yes ,Join Dreams SchoolDreams School ““CampaignCampaign for Female Educationfor Female Education”” Help us in bringing a change in a girl life,Help us in bringing a change in a girl life, becausebecause ““When someone takes awayWhen someone takes away your pens you realize quite howyour pens you realize quite how important education isimportant education is”.”. Just Click on any advertisement on the page,Just Click on any advertisement on the page, your one click can make her smileyour one click can make her smile.. We our doing our part & u ?We our doing our part & u ? Eliminate InequalityEliminate Inequality ““Not WomenNot Women””