Econ 205 - Slides from Lecture 1Joel SobelAugust 23, 2010Econ 205 SobelWarningI can’t start without assuming that something is “commonknowledge.” You can find basic definitions of Sets and SetOperations (Union, Intersection, . . . ) in the lecture notes.Econ 205 SobelFunctionsDefinitionA function f from a set X to a set Y is a specification (amapping) that assigns to each element of X exactly one element ofY . Typically we express that f is such a mapping by writingf :X → Y . The set X (the points one “plugs into” the function) iscalled the domain of the function, and the set Y (the items thatone can get out of the function) is called the function’s codomainor range.We write f (x) as the point in Y that the function associates withx ∈ X .Econ 205 SobelDefinitionThe image of a function f from a set X to a set Y is{y ∈ Y : f (x) = y for some x ∈ X }. We denote this set by f (X ).DefinitionThe inverse image of a set W ⊂ Y asf−1(W ) ≡ {x ∈ X | f (x) ∈ W }.This is the set of points in X that map to points in W .Note that the inverse image does not necessarily define a functionfrom Y to X , because it could be that there are two distinctelements of X that map to the same point in Y .Econ 205 SobelMore DefinitionsDefinitionA function f :X → Y is onto if f (X ) = Y .DefinitionConsider a function f :X → Y . This function is said to beone-to-one if f (x) = f (x0) implies that x = x0. An equivalentcondition is that for every two points x, x0∈ X such that x 6= x0, itis the case that f (x) 6= f (x0).DefinitionA function f :X → Y is called invertible if the inverse imagemapping is a function from Y to X . Then, for every y ∈ Y ,f−1(y) is defined to be the point x ∈ X for which f (x) = y.Econ 205 SobelTheoremA function is invertible if and only if it is both one-to-one and onto.Econ 205 SobelAlgebra of FunctionsDefinitionSuppose we have functions g : X → Y and f : Y → Z. Then thecomposition function f ◦ g is a mapping from X to Z (that is,f ◦ g :: X → Z ). Writing h = f ◦ g, this function is defined byh(x) ≡ f (g (x)) for every x ∈ X.Econ 205 SobelWhen the range has special structure (as it will in almost allapplications), there are other algebraic operations:DefinitionIf f , g : X → Y , then we can form new functions from X → YIThe sum (denoted f + g) and defined by(f + g)(x) = f (x) + g(x).IMultiplication by a constant (denoted λf ) and defined by(λf )(x) = λf (x).IThe product (denoted fg) and defined by (fg)(x) = f (x)g(x).IThe quotient (denoted f /g) and defined by(f /g)(x) = f (x)/g(x).These definitions make sense exactly when it is possible to add,multiply by a constant, multiply, or divide elements of Y .When is that possible?Econ 205 SobelThe Real LineThe set of real numbers, denoted by R. A class in “real analysis”would construct the real line from more basic sets – first integers,then ratios of integers (called rational numbers), then (in oneconstruction) limits of rational numbers. We will skip this.For a, b ∈ R where a ≤ bIClosed Interval: [a, b] ≡ {x : a ≤ x ≤ b}IHalf-open interval: (a, b] ≡ {x : a < x ≤ b} or[a, b) ≡ {x : a ≤ x < b}IOpen interval: (a, b) ≡ {x :| a < x < b}Can have a = −∞ or b = ∞.For example, (−∞, b) ≡ {x : x < b}.Econ 205 SobelBoundsDefinitionTake X ⊂ R. a ∈ R is an upper bound for X ifa ≥ x, ∀x ∈ Xb ∈ R is a lower bound for X ifb ≤ x, ∀x ∈ XThe set X is bounded” if “X is bounded from above” and “X isbounded from below.”Econ 205 SobelDefinitiona ∈ R is a least upper bound of X , or the supremum of X , ifa ≥ x, ∀x ∈ X (a is an upper bound), and if a0is also an upperbound then a0≥ a. We write a = sup X .Econ 205 SobelDefinitionb ∈ R is a greatest lower bound of X , or the infimum of X , ifb ≤ x, ∀x ∈ X (b is an lower bound), and if b0is also an lowerbound then b0≤ b. We writeb = inf Xsup and inf need not be in set ((0, 1)).TheoremAny set X ⊂ R, (X 6= φ), that has an upper bound, has a leastupper bound (sup).Econ 205 SobelDefinitionmax X ≡ a number a such thata ∈ X and a ≥ x, ∀x ∈ Xandmin X ≡ a number b such that b ∈ X and b ≤ x, ∀x ∈ XMax and min do not always exist even if the set is bounded, butthe sup and the inf do always exist if the set is bounded. Ifsup X ∈ X, then max X exists and is equal to sup X .Econ 205 SobelSimple ResultTheoremIf max X exists thenmax X = sup XLet a ≡ max XWe will demonstrate that a is sup X .To do this we must show that:(i) a is an upper bound(ii) Every other upper bound a0satisfies a0≥ aso(i) a is an upper bound on X since by the definition of maxa ≥ x, ∀x ∈ XEcon 205 Sobel(ii) Consider any other upper bound a0of the set X such thata06= a.Since a0is an upper bound, and a ∈ X (by the fact that a = maxX ), we must have a0> a.Econ 205 SobelSequencesLet P denote the positive integers.DefinitionA sequence is a function f from P to R.f : P −→ RIf f (n) = an, for n ∈ P, we denote the sequence f by the symbol{an}∞n=1, or sometimes by {a1, a2, a3, . . . }. The values of f , thatis, the elements an, are called the terms of the sequence.Econ 205 SobelExamples1. an= n, {an}∞n=1= {1, 2, 3, . . . }2. a1= 1, an+1= an+ 2, ∀n > 1, {an}∞n=1= {1, 3, 5, . . . }3. an= (−1)n, n ≥ 1,{an}∞n=1= {−1, 1, −1, 1, . . . }Econ 205 SobelLimits of SequencesDefinition[Convergent Sequence]A sequence {an} is said to converge, if there is a point a ∈ R suchthat for every ε > 0 ∃N ∈ P such that if n ≥ N, then | an−a |< ε.In this case we also say that {an} converges to a and we writean−→ aorlimn→∞an= aIf {an} does not converge, it is said to diverge.Econ 205 SobelPropertiesTheoremLet {an} be a sequence in R.If b ∈ R, b0∈ R, and if {an} converges to b and to b0, thenb = b0Proof.Assume, in order to obtain a contradiction, that b > b0Let ε =12(b − b0) > 0.By definition, there is N and N0such that∀n ≥ N| b − an|< εand ∀n ≥ N0| b0− an|< εThis is …