DOCSLIB.ORG

Give the Definition of Matrix-Matrix Multiplication (A Times B)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Give the Definition of Matrix-Matrix Multiplication (A Times B) Quiz Give the definition of matrix-matrix multiplication (A times B) in which one of the matrices (which?) is broken into rows and columns (which?). In your definition, specify how the output matrix is obtained from the input matrices A and B. Illustrate your definition with an explicit numeric example. Your definition should not explicitly mention dot-products or linear combinations. What is the relation between matrix-matrix multiplication and function composition? Illustrate with an explicit numeric example. Review of concept of linear transformation A function f : where and are vector spaces is a linear transformation if V−!W V W Property L1: For every vector v in and every scalar ↵ in F, f (↵ v)=↵ f (v) V Property L2: For every two vectors u and v in , f (u + v)=f (u)+f (v) V Things to note: Most common example is: Given R C matrix M, ⇥ function is f : FC FR with rule f (x)=Mx ! We proved that such a function is a linear transformation. But this is not only kind of linear transformation: C R I and can be arbitrary vector spaces (not just F or F ) V W I The function f can be described in another way, e.g. “rotation by angle ⇡/3withinthe plane Span [1, 2, 3], [9, 1, 7] ” { } Nevertheless, it’s true that if domain of f is a subspace of FC and co-domain is a subspace of FR and f is a linear transformation then f can be represented as matrix-vector multiplication. Which functions are linear transformations? Define s([x, y]) = stretching by two in horizontal direction Property L1: s(v1 + v2)=s(v1)+s(v2) Property L2: s(↵ v)=↵ s(v) v1 and s(v1) (1,1) (2,1) Since the function s( ) satisfies Properties L1 and L2, it · is a linear transformation. Similarly can show rotation by ✓ degrees is a linear v2 and s(v2) (1,2) transformation. (2,2) What about translation? t([x, y]) = [x, y]+[1, 2] + This function violates Property L1. For example: t([4, 5] + [2, 1]) = t([6, 4]) =[7, 6] − but (2,3) (4,3) v1+v2 and s(v1+v2) t([4, 5]) + t([2, 1]) = [5, 7] + [3, 1] =[8, 8] − Since t( ) violates Property L1 for at · least one input, it is not alinear transformation. Can similarly show that t( ) does not · satisfy Property L2. Which functions are linear transformations? Define s([x, y]) = stretching by two in horizontal direction Property L1: s(v1 + v2)=s(v1)+s(v2) × 1.5 Property L2: s(↵ v)=↵ s(v) 1.5 v1 and s(1.5 v1) Since the function s( ) satisfies Properties L1 and L2, it (1.5,1.5) (3,1.5) · is a linear transformation. Similarly can show rotation by ✓ degrees is a linear Since t( ) violates Property L1 for at · transformation. least one input, it is not alinear transformation. What about translation? t([x, y]) = [x, y]+[1, 2] Can similarly show that t( ) does not · This function violates Property L1. For example: satisfy Property L2. t([4, 5] + [2, 1]) = t([6, 4]) =[7, 6] − but t([4, 5]) + t([2, 1]) = [5, 7] + [3, 1] =[8, 8] − Linear transformations: Pushing linear combinations through the function Defining properties of linear transformations: Property L1: f (↵ v)=↵ f (v) Property L2: f (u + v)=f (u)+f (v) Proposition: For a linear transformation f , for any vectors v1,...,vn in the domain of f and any scalars ↵1,...,↵n, f (↵ v + + ↵ v )=↵ f (v )+ + ↵ f (v ) 1 1 ··· n n 1 1 ··· n n “Proof”: Consider the case of n = 2. f (↵1 v1 + ↵2 v2) = f (↵1 v1)+f (↵2 v2) by Property L2 = ↵1 f (v1)+↵2 f (v2) by Property L1 Proof for general n is similar. QED linear transformations: Pushing linear combinations through the function Proposition: For a linear transformation f , f (↵ v + + ↵ v )=↵ f (v )+ + ↵ f (v ) 1 1 ··· n n 1 1 ··· n n 12 Example: f (x)= x 34 ⇤ Verify that f (10 [1, 1] + 20 [1, 0]) = 10 f ([1, 1]) + 20 f ([1, 0]) − − 12 10 [1, 1] + 20 [1, 0] 34 − ! 12 12 12 10 [1, 1] + 20 [1, 0] = [10, 10] + [20, 0] 34 ⇤ − ! 34 ⇤ ! 34 − ! = 10 ([1, 3] [2, 4]) + 20 (1[1, 3]) 12 − = [30, 10] = 10 [ 1, 1] + 20 [1, 3] 34 − − − =[10, 10] + [20, 60] = 30 [1, 3] 10[2, 4] − − − = [10, 50] = [30, 90] [20, 40] − = [10, 50] From function to matrix, revisited We saw a method to derive a matrix from a function: Given a function f : Rn Rm,wewantamatrixM such that f (x)=M x.... ! ⇤ I Plug in the standard generators e1 =[1, 0,...,0, 0],...,en =[0,...,0, 1] I Column i of M is f (ei ). This works correctly whenever such a matrix M really exists: Proof: If there is such a matrix then f is linear: I (Property L1) f (↵ v)=↵ f (v)and I (Property L2) f (u + v)=f (u)+f (v) n Let v =[↵1,...,↵n]beanyvectorinR . We can write v in terms of the standard generators. v = ↵ e + + ↵ e 1 1 ··· n n so f (v)=f (↵ e + + ↵ e ) 1 1 ··· n n = ↵ f (e )+ + ↵ f (e ) 1 1 ··· n n = ↵ (column 1 of M)+ + ↵ (columnn of M) = M v QED 1 ··· n ⇤ Alineartransformationmapszerovectortozerovector Lemma: If f : is a linear transformation then f maps the zero vector of to the zero U−!V U vector of . V Proof: Let 0 denote the zero vector of ,andlet0 denote the zero vector of . U V V f (0)=f (0 + 0)=f (0)+f (0) Subtracting f (0) from both sides, we obtain 0 = f (0) V QED linear transformations and zero vectors: Kernel Definition: Kernel of a linear transformation f is v : f (v)=0 { } Written Ker f For a function f (x)=M x, ⇤ Ker f =NullM Kernel and one-to-one One-to-One Lemma: A linear transformation is one-to-one if and only if its kernel is a trivial vector space. Proof: Let f : be a linear transformation. We prove two directions. U−!V I Suppose Ker f contains some nonzero vector u,sof (u)=0 . V Because a linear transformation maps zero to zero, f (0)=0 as well, V so f is not one-to-one. I Suppose Ker f = 0 . { } Let v1, v2 be any vectors such that f (v1)=f (v2). Then f (v1) f (v2)=0 − V so, by linearity, f (v1 v2)=0 , − V so v v Ker f . 1 − 2 2 Since Ker f consists solely of 0, it follows that v v = 0,sov = v . 1 − 2 1 2 QED Kernel and one-to-one One-to-One Lemma A linear transformation is one-to-one if and only if its kernel is a trivial vector space. Define the function f (x)=A x. ⇤ If Ker f is trivial (i.e. if Null A is trivial) then a vector b is the image under f of at most one vector. That is, at most one vector u such that A u = b ⇤ That is, the solution set of A x = b has at most one vector. ⇤ linear transformations that are onto? Question: How can we tell if a linear transformation is onto? Recall: for a function f : ,theimage of f is the set of all images of elements of the V−!W domain: f (v):v { 2V} (You might know it as the “range” but we avoid that word here.) The image of function f is written Im f “Is function f is onto?” same as “is Im f = co-domain of f ?” Example: Lights Out Define f ([↵1,↵2,↵3,↵4]) = 2 •• •• • • 3 [↵1,↵2,↵3,↵4] ⇤ • • •• •• 6 7 6 7 Im f is set of configurations4 for which 2 2 Lights Out can be solved,5 ⇥ so “f is onto” means “2 2 Lights Out can be solved for every configuration” ⇥ Can 2 2 Lights Out be solved for every configuration? What about 5 5? ⇥ ⇥ Each of these questions amounts to asking whether a certain function is onto. Linear transformations that are onto? “Is function f is onto?” same as “is Im f = co-domain of f ?” First step in understanding how to tell if a linear transformation f is onto: I study the image of f Proposition: The image of a linear transformation f : is a vector space V−!W The image of a linear transformation is a vector space Proposition: The image of a linear transformation f : is a vector space V−!W Recall: aset of vectors is a vector space if U V1: contains a zero vector, U V2: for every vector w in and every scalar ↵, the vector ↵ w is in U U V3: for every pair of vectors w and w in , the vector w + w is in 1 2 U 1 2 U Proof: V1: Since the domain contains a zero vector 0 and f (0 )=0 , the image of f includes V V V W 0 . This proves Property V1. W V2: Suppose some vector w is in the image of f . That means there is some vector v in the domain that maps to w: f (v)=w. V By Property L1, for any scalar ↵, f (↵ v)=↵ f (v)=↵ w so ↵ w is in the image. This proves Property V2. V3: Suppose vectors w1 and w2 are in the image of f . That is, there are vectors v1 and v2 in the domain such that f (v1)=w1 and f (v2)=w2.
Load more

Recommended publications
  • Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: at the End of Class on Tuesday, March 26Th
    Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: At the end of class on Tuesday, March 26th More on Functions: Definition 5.1.7: Let A, B, C, and D be sets. Let f : A B and g : C D be functions. Then f = g if: → → A = C and B = D • For all x A, f(x)= g(x). • ∈ Intuitively speaking, this definition tells us that a function is determined by its underlying correspondence, not its specific formula or rule. Another way to think of this is that a function is determined by the set of points that occur on its graph. 1. Give a specific example of two functions that are defined by different rules (or formulas) but that are equal as functions. 2. Consider the functions f(x)= x and g(x)= √x2. Find: (a) A domain for which these functions are equal. (b) A domain for which these functions are not equal. Definition 5.1.9 Let X be a set. The identity function on X is the function IX : X X defined by, for all x X, → ∈ IX (x)= x. 3. Let f(x)= x cos(2πx). Prove that f(x) is the identity function when X = Z but not when X = R. Definition 5.1.10 Let n Z with n 0, and let a0,a1, ,an R such that an = 0. The function p : R R is a ∈ ≥ ··· ∈ 6 n n−1 → polynomial of degree n with real coefficients a0,a1, ,an if for all x R, p(x)= anx +an−1x + +a1x+a0.
    [Show full text]
  • 1 Monoids and Groups
    1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M × M ! M; (x; y) 7! x · y such that (i) (x · y) · z = x · (y · z) 8x; y; z 2 M (associativity); (ii) 9e 2 M such that x · e = e · x = x for all x 2 M (e = the identity element of M). 1.2 Examples. 1) Z with addition of integers (e = 0) 2) Z with multiplication of integers (e = 1) 3) Mn(R) = fthe set of all n × n matrices with coefficients in Rg with ma- trix multiplication (e = I = the identity matrix) 4) U = any set P (U) := fthe set of all subsets of Ug P (U) is a monoid with A · B := A [ B and e = ?. 5) Let U = any set F (U) := fthe set of all functions f : U ! Ug F (U) is a monoid with multiplication given by composition of functions (e = idU = the identity function). 1.3 Definition. A monoid is commutative if x · y = y · x for all x; y 2 M. 1.4 Example. Monoids 1), 2), 4) in 1.2 are commutative; 3), 5) are not. 1 1.5 Note. Associativity implies that for x1; : : : ; xk 2 M the expression x1 · x2 ····· xk has the same value regardless how we place parentheses within it; e.g.: (x1 · x2) · (x3 · x4) = ((x1 · x2) · x3) · x4 = x1 · ((x2 · x3) · x4) etc. 1.6 Note. A monoid has only one identity element: if e; e0 2 M are identity elements then e = e · e0 = e0 1.7 Definition.
    [Show full text]
  • Lecture 3 Partially Ordered Sets
    Lecture 3 Partially ordered sets Now for something totally different. In life, useful sets are rarely “just” sets. They often come with additional structure. For example, R isn’t useful just because it’s some set. It’s useful because we can add its elements, multiply its elements, and even compare its elements. If A is, for example, the set of all bananas and apples on earth, it’s not as natural to do any of these things with elements of A. Today, we’ll talk about a structure called a partial order. 3.1 Preliminaries First, let me remind you of something called the Cartesian product, or direct product of sets. Definition 3.1.1. Let A and B be sets. Then the Cartesian product of A and B is defined to be the set consisting of all pairs (a, b) where a A and œ b B. We denote the Cartesian product by œ A B. ◊ Example 3.1.2. R2 is the Cartesian product R R. ◊ Example 3.1.3. Let T be the set of all t-shirts in Hiro’s closet, and S the set of all shorts in Hiro’s closet. Then S T represents the collection of all ◊ possible outfits Hiro is willing to consider in August. (I.e., Hiro is willing to consider putting on any of his shorts together with any of his t-shirts.) 23 24 LECTURE 3. PARTIALLY ORDERED SETS Note that we can “repeat” the Cartesian product operation, or apply it many times at once. Example 3.1.4.
    [Show full text]
  • Lecture 19: Alternating Group
    MATH 433 Applied Algebra Lecture 19: Alternating group. Abstract groups. Sign of a permutation Theorem 1 For any n ≥ 2 there exists a unique function sgn : S(n) → {−1, 1} such that • sgn(πσ)= sgn(π) sgn(σ) for all π, σ ∈ S(n), • sgn(τ)= −1 for any transposition τ in S(n). A permutation π is called even if it is a product of an even number of transpositions, and odd if it is a product of an odd number of transpositions. It turns out that π is even if sgn(π)=1 and odd if sgn(π)= −1. Theorem 2 (i) sgn(πσ)= sgn(π) sgn(σ) for any π, σ ∈ S(n). (ii) sgn(π−1)= sgn(π) for any π ∈ S(n). (iii) sgn(id) = 1. (iv) sgn(τ)= −1 for any transposition τ. (v) sgn(σ)=(−1)r−1 for any cycle σ of length r. Alternating group Given an integer n ≥ 2, the alternating group on n symbols, denoted An or A(n), is the set of all even permutations in the symmetric group S(n). Theorem (i) For any two permutations π, σ ∈ A(n), the product πσ is also in A(n). (ii) The identity function id is in A(n). (iii) For any permutation π ∈ A(n), the inverse π−1 is in A(n). In other words, the product of even permutations is even, the identity function is an even permutation, and the inverse of an even permutation is even. Theorem The alternating group A(n) has n!/2 elements.
    [Show full text]
  • Theoretical Probability and Statistics
    Theoretical Probability and Statistics Charles J. Geyer Copyright 1998, 1999, 2000, 2001, 2008 by Charles J. Geyer September 5, 2008 Contents 1 Finite Probability Models 1 1.1 Probability Mass Functions . 1 1.2 Events and Measures . 5 1.3 Random Variables and Expectation . 6 A Greek Letters 9 B Sets and Functions 11 B.1 Sets . 11 B.2 Intervals . 12 B.3 Functions . 12 C Summary of Brand-Name Distributions 14 C.1 The Discrete Uniform Distribution . 14 C.2 The Bernoulli Distribution . 14 i Chapter 1 Finite Probability Models The fundamental idea in probability theory is a probability model, also called a probability distribution. Probability models can be specified in several differ- ent ways • probability mass function (PMF), • probability density function (PDF), • distribution function (DF), • probability measure, and • function mapping from one probability model to another. We will meet probability mass functions and probability measures in this chap- ter, and will meet others later. The terms “probability model” and “probability distribution” don’t indicate how the model is specified, just that it is specified somehow. 1.1 Probability Mass Functions A probability mass function (PMF) is a function pr Ω −−−−→ R which satisfies the following conditions: its values are nonnegative pr(ω) ≥ 0, ω ∈ Ω (1.1a) and sum to one X pr(ω) = 1. (1.1b) ω∈Ω The domain Ω of the PMF is called the sample space of the probability model. The sample space is required to be nonempty in order that (1.1b) make sense. 1 CHAPTER 1. FINITE PROBABILITY MODELS 2 In this chapter we require all sample spaces to be finite sets.
    [Show full text]
  • Inverse Monoids Associated with the Complexity Class NP
    Inverse monoids associated with the complexity class NP J.C. Birget 6 March 2017 Abstract We study the P versus NP problem through properties of functions and monoids, continuing the work of [3]. Here we consider inverse monoids whose properties and relationships determine whether P is different from NP, or whether injective one-way functions (with respect to worst-case complexity) exist. 1 Introduction We give a few definitions before motivating the monoid approach to the P versus NP problem. Some of these notions appeared already in [3]. By function we mean a partial function A∗ → A∗, where A is a finite alphabet (usually, A = {0, 1}), and A∗ denotes the set of all finite strings over A. Let Dom(f) (⊆ A∗) denote the domain of f, i.e., {x ∈ A∗ : f(x) is defined}; and let Im(f) (⊆ A∗) denote the image (or range) of f, i.e., {f(x) : x ∈ ∗ ∗ Dom(f)}. The length of x ∈ A is denoted by |x|. The restriction of f to X ⊆ A is denoted by f|X , and the identity function on X is denoted by idX or id|X . Definition 1.1 (inverse, co-inverse, mutual inverse) A function f ′ is an inverse of a function f iff f ◦ f ′ ◦ f = f. In that case we also say that f is a co-inverse of f ′. If f ′ is both an inverse and a co-inverse of f, we say that f ′ is a mutual inverse of f, or that f ′ and f are mutual inverses (of each other).1 It is easy to see that f ′ is an inverse of f iff for every y ∈ Im(f): f ′(y) is defined and f ′(y) ∈ f −1(y).
    [Show full text]
  • Math 131: Introduction to Topology 1
    Math 131: Introduction to Topology 1 Professor Denis Auroux Fall, 2019 Contents 9/4/2019 - Introduction, Metric Spaces, Basic Notions3 9/9/2019 - Topological Spaces, Bases9 9/11/2019 - Subspaces, Products, Continuity 15 9/16/2019 - Continuity, Homeomorphisms, Limit Points 21 9/18/2019 - Sequences, Limits, Products 26 9/23/2019 - More Product Topologies, Connectedness 32 9/25/2019 - Connectedness, Path Connectedness 37 9/30/2019 - Compactness 42 10/2/2019 - Compactness, Uncountability, Metric Spaces 45 10/7/2019 - Compactness, Limit Points, Sequences 49 10/9/2019 - Compactifications and Local Compactness 53 10/16/2019 - Countability, Separability, and Normal Spaces 57 10/21/2019 - Urysohn's Lemma and the Metrization Theorem 61 1 Please email Beckham Myers at [email protected] with any corrections, questions, or comments. Any mistakes or errors are mine. 10/23/2019 - Category Theory, Paths, Homotopy 64 10/28/2019 - The Fundamental Group(oid) 70 10/30/2019 - Covering Spaces, Path Lifting 75 11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80 11/6/2019 - The Brouwer Fixed Point Theorem 85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88 11/13/2019 - Deformation Retracts and Homotopy Equivalence 91 11/18/2019 - Computing the Fundamental Group 95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99 11/25/2019 - Universal Covering Spaces, Free Groups 104 12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109 2 9/4/2019 - Introduction, Metric Spaces, Basic Notions The instructor for this course is Professor Denis Auroux. His email is [email protected] and his office is SC539.
    [Show full text]
  • 19. Automorphism Group of S Definition-Lemma 19.1. Let G Be A
    19. Automorphism group of Sn Definition-Lemma 19.1. Let G be a group. The automorphism group of G, denoted Aut(G), is the subgroup of A(Sn) of all automorphisms of G. Proof. We check that Aut(G) is closed under products and inverses. Suppose that φ and 2 Aut(G). Let ξ = φ ◦ . If g and h 2 G then ξ(gh) = (φ ◦ )(gh) = φ( (gh)) = φ( (g) (h)) = φ( (g))φ( (h)) = (φ ◦ )(g)(φ ◦ )(h) = ξ(g)ξ(h): Thus ξ = φ◦ is a group homomorphism. Thus Aut(G) is closed under products. Now let ξ = φ−1. If g and h 2 G then we can find g0 and h0 such that g = φ(g0) and h = φ(h0). It follows that ξ(gh) = ξ(φ(g0)φ(h0)) = ξ(φ(g0h0)) = g0h0 = ξ(g)ξ(h): Thus ξ = φ−1 is a group homomorphism. Thus Aut(G) is closed under inverses. Lemma 19.2. Let G be a group and let a 2 G. φa is the automorphism of G given by conjugation by a, φ(g) = aga−1. If a and b 2 G then φab = φaφb: Proof. Both sides are functions from G to G. We just need to check that they have the same effect on any element g of G: (φa ◦ φb)(g) = φa(φb(g)) −1 = φa(bgb ) = a(bgb−1)a−1 = (ab)g(ab)−1 = φab(g): 1 Definition-Lemma 19.3. We say that an automorphism φ of G is inner if φ = φa for some a.
    [Show full text]
  • Degree 2 Transformation Semigroups As Continuous Maps on Graphs
    Degree 2 Transformation Semigroups as Continuous Maps on Graphs: Foundations and Structure Stuart Margolis, John Rhodes May 7, 2020 2010 Mathematics Subject Classification: 20M10, 20M20, 20M30 Keywords: degree of a transformation semigroup, complexity of semigroups ABSTRACT We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn-Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter mapping image and relate this to their graph of fibers. 1 Introduction A transformation semigroup (ts) is a pair X = (Q,S), where Q is a finite set and S is a subsemigroup of the monoid P FR(Q) of all partial functions on Q, where composition is from left to right. A transformation monoid (tm) is a transformation semigroup X = (Q,S) such that the identity function on Q is in S. Equivalently, a ts is a faithful right action of a semigroup S on a finite set Q by partial functions and a tm is a faithful right action of a monoid on Q. If f : Q → Q is a partial function, then arXiv:2005.02606v1 [math.GR] 6 May 2020 ker(f) is the equivalence relation on the domain of f that identifies two elements x,y if xf = yf.
    [Show full text]
  • Chapter 8 Ordered Sets
    Chapter VIII Ordered Sets, Ordinals and Transfinite Methods 1. Introduction In this chapter, we will look at certain kinds of ordered sets. If a set \ is ordered in a reasonable way, then there is a natural way to define an “order topology” on \. Most interesting (for our purposes) will be ordered sets that satisfy a very strong ordering condition: that every nonempty subset contains a smallest element. Such sets are called well-ordered. The most familiar example of a well-ordered set is and it is the well-ordering property that lets us do mathematical induction in In this chapter we will see “longer” well ordered sets and these will give us a new proof method called “transfinite induction.” But we begin with something simpler. 2. Partially Ordered Sets Recall that a relation V\ on a set is a subset of \‚\ (see Definition I.5.2 ). If ÐBßCÑ−V, we write BVCÞ An “order” on a set \ is refers to a relation on \ that satisfies some additional conditions. Order relations are usually denoted by symbols such asŸ¡ß£ , , or . Definition 2.1 A relation V\ on is called: transitive ifÀ a +ß ,ß - − \ Ð+V, and ,V-Ñ Ê +V-Þ reflexive ifÀa+−\+V+ antisymmetric ifÀ a +ß , − \ Ð+V, and ,V+ Ñ Ê Ð+ œ ,Ñ symmetric ifÀ a +ß , − \ +V, Í ,V+ (that is, the set V is “symmetric” with respect to thediagonal ? œÖÐBßBÑÀB−\ש\‚\). Example 2.2 1) The relation “œ\ ” on a set is transitive, reflexive, symmetric, and antisymmetric. Viewed as a subset of \‚\, the relation “ œ ” is the diagonal set ? œÖÐBßBÑÀB−\×Þ 2) In ‘, the usual order relation is transitive and antisymmetric, but not reflexive or symmetric.
    [Show full text]
  • Math 3310: Intro to Proofs
    Math 3310: Intro to Proofs Stephen Pe~na Contents 1 Introduction: Why proofs?2 2 A Primer: Propositional Logic3 2.1 The Language.................................3 2.2 Quantifiers...................................4 2.3 Truth Tables.................................5 2.4 Conditional and Biconditional........................8 3 Sets and Functions9 3.1 The Basics of Sets..............................9 3.2 Functions................................... 12 3.3 Injective and Surjective Functions...................... 14 4 Analysis 17 4.1 Continuity................................... 17 4.2 Continuity of Polynomials.......................... 19 4.3 Further Reading................................ 20 5 Algebra 21 5.1 Groups..................................... 21 5.2 Homomorphisms............................... 24 5.3 Equivalence Relations............................. 25 5.4 Subgroups and Normal Subgroups...................... 28 6 Topology 30 6.1 Topological Spaces.............................. 30 6.2 Continuous Maps of Topological Spaces................... 31 A Countable vs Uncountable 32 B Proof by Induction 32 1 1 Introduction: Why proofs? Many of us are familiar with a myriad of mathematical statements, and we may even be convinced of their veracity. Beliefs aside, how can we convince ourselves a statement is genuinely true? Most importantly, how can you convince your fellow mathematicians that something is true? The answer is by presenting a coherent mathematical argument involving already established facts. Such a coherent mathematical argument is called a proof. Example 1.1. Suppose we know for a fact that 1+1=2. We are tasked with proving the following: 2 + 1 = 3: One way we could proceed is as follows: Proposition 1.1. 2+1=3. Proof. We have that 1+1=2. Note that 3 = 1 + 1 + 1. Substituting in 1 + 1 = 2, we have 3 = 2 + 1 = 1 + 2: There are a few things to note about the above proof.
    [Show full text]
  • On Some Properties of the Free Monoids with Applications to Automata Theory*
    JOURNAL OF COMPUTER AND SYSTEM SCIENCES: 1, 13%154 (1967) On Some Properties of the Free Monoids with Applications to Automata Theory* YEHOSHAFAT GIVE'ON The Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138 INTRODUCTION Let T be any set, and denote by T* the free monoid generated by T. By definition, T* is characterized by the property that, for any function f from the set T into the carrier of any monoid M, there exists a unique monoid homornorphism f* : T* -+ M whose underlying function is an extension of the function f. This property of the free monoids is very useful in many applications where free monoids are encountered. Another property of the free monoids follows immediately from this definition. Namely, for any homomorphism h : T*-+ M 2 and any surjective homomorphism f : M 1 -~ M 2 there exists at least one homomorphism g : T* ~ M 1 such that Ylhm ~ M l ~ M a f commutes. This paper will be concerned with the following issues: (i) The above-mentioned derived property of the free monoids characterizes the free monoids in the category M of all monoids. This property should not be confused with either the defining property of the free monoids or with the property of the projective objects in 19I. On the other hand, our method of proof yields the analogous characterization of the free Abelian monoids in the category of Abelian monoids. (ii) Since every monoid is a homomorphic image of a free monoid, the above- derived property leads us to consider idempotent endomorphisms of free monoids.
    [Show full text]