DOCSLIB.ORG

CSE 1400 Applied Discrete Mathematics Relations and Functions Department of Computer Sciences College of Engineering Florida Tech

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CSE 1400 Applied Discrete Mathematics Relations and Functions Department of Computer Sciences College of Engineering Florida Tech CSE 1400 Applied Discrete Mathematics Relations and Functions Department of Computer Sciences College of Engineering Florida Tech Spring 2012 Problems labeled ! are important and must be Problems on Relations mastered. are harder and cover interest- Background ing advanced material that is not properly part of this course. A (binary) relation is a set of ordered pairs are tangential to the course f(x, y) : x ∼ yg but cover material that is useful elsewhere. where ∼ is a relational symbol, for example, equal (=), less than (<), divides (j), congruence mod n (≡ (mod n)), subset (⊆), et cetera. Two important types of relations are equivalences and (partial) orders: • Equivalence relations are – reflexive (8x)(x ∼ x), – symmetric (8x, y)(x ∼ y ! y ∼ x), and – transitive (8x, y, z)((x ∼ y) ^ (y ∼ z) ! x ∼ z). • Partial orders are reflexive, transitive, and antisymmetric (8x, y)((x ∼ y) ^ (y ∼ x) ! x = y) or by contraposition (8x, y)(x 6= y ! (x 6∼ y) _ (y 6∼ x)) 1. ! (Recognize whether several important relations are reflexive or not.) Which of the following state- ments are True and which are False? (a) (8x 2 R)(x = x). Answer: This is True. Equality is a reflexive relation. (b) (8x 2 R)(x 6= x). Answer: This is False. Not equal is not a reflexive relation. (c) (8x 2 R)(x < x). Answer: This is False. Less than is not a reflexive relation. (d) (8x 2 R)(x ≥ x). Answer: This is True. Greater than or equal is a reflexive relation. (e) (8a 2 N)(a j a). Answer: This is True. Divides is a reflexive relation. (f) (8X 2 2U)(X ⊆ X). Answer: This is True. Subset is a reflexive relation. cse 1400 applied discrete mathematics relations and functions 2 (g) Let n 2 N, n > 1 be fixed. (8a 2 Z)(a ≡ a (mod n)). Answer: This is True. Congruence mod n is a reflexive relation. (h) (8a 2 Z)(gcd(a, a) = 1) Answer: This is False. The greatest common divisor of a and a is jaj, which is most often not equal to 1. That is, relatively prime is not a reflexive relation. 2. ! (Recognize whether several important relations are symmetric or not.) Which of the following state- ments are True and which are False? (a) (8x, y 2 R)((x = y) ! (y = x)). Answer: This is True. Equality is symmetric. (b) (8x, y 2 R)((x 6= y) ! (y 6= x)). Answer: This is True. Not equal is symmetric (c) (8x, y 2 R)(x < y ! y < x). Answer: This is False. Less than is not symmetric. (d) (8x, y 2 R)(x ≥ y ! y ≥ x). Answer: This is False. Greater than or equal is not symmetric. (e) (8a, b 2 N)(a j b ! b j a). Answer: This is False. Divides is not symmetric. (f) (8X, Y 2 2U)(X ⊆ Y ! Y ⊆ X). Answer: This is False. Subset is not symmetric. (g) Let n 2 N, n > 1 be fixed. (8a, b 2 Z)(a ≡ b (mod n) ! b ≡ a (mod n)). Answer: This is True. Congruence mod n is symmetric. (h) (8a, b 2 Z)(gcd(a, b) = 1 ! gcd(b, a) = 1). Answer: This is True. Relatively prime is symmetric. 3. ! (Recognize whether several important relations are antisymmetric or not.) Which of the following statements are True and which are False? (a) (8x, y 2 R)(((x = y) ^ (y = x)) ! (x = y)). Answer: This is True. Equality is antisymmetric. (b) (8x, y 2 R)((x 6= y ^ y 6= x) ! (x = y)). Answer: This is False. Not equal in not antisymmetric (c) (8x, y 2 R)((x < y ^ y < x) ! (x = y)). Answer: This is True. It is True by virtue of its logical form: p ! q where the premise p = (x < y ^ y < x) is always False which makes the conditional always True. (d) (8x, y 2 R)(x ≥ y ^ y ≥ x ! x = y). Answer: This is True. Greater than or equal is antisymmetric. (e) (8a, b 2 N)(a j b ^ b j a ! a = b). Answer: This is True. Divides is antisymmetric. (f) (8X, Y 2 2U)((X ⊆ Y) ^ (Y ⊆ X) ! X = Y). Answer: This is True. Subset is antisymmetric. (g) Let n 2 N, n > 1 be fixed. (8a, b 2 Z)(a ≡ b (mod n) ^ b ≡ a (mod n) ! a = b). Answer: This is False. Congruence mod n is not antisymmetric. cse 1400 applied discrete mathematics relations and functions 3 (h) (8a, b 2 Z)((gcd(a, b) = 1) ^ (gcd(b, a) = 1) ! a = b). Answer: This is False. Relatively prime is not antisymmetric. 4. ! (Recognize whether several important relations are transitive or not.) Which of the following state- ments are True and which are False? (a) (8x, y, z 2 R)(((x = y) ^ (y = z)) ! (x = z)). Answer: This is True. Equality is transitive. (b) (8x, y, z 2 R)((x 6= y ^ y 6= z) ! (x 6= z)). Answer: This is False. Not equal is not transitive. A counterexample to the statement that not equal is transitive is: x = 0, y = 1 and z = 0. (c) (8x, y, z 2 R)((x < y ^ y < z) ! (x < z)). Answer: This is True. Less than is transitive. (d) (8x, y, z 2 R)(x ≥ y ^ y ≥ z ! x ≥ z). Answer: This is True. Greater than or equal is transitive. (e) (8a, b, c 2 N)(a j b ^ b j c ! a j c). Answer: This is True. If a j b, then there is a natural number d such that ad = b. And, If b j c, then there is a natural number e such that be = c. Therefore a(de) = (ad)e = be = c and a divides c. Divides is transitive. (f) (8X, Y, Z 2 2U)((X ⊆ Y) ^ (Y ⊆ Z) ! X ⊆ Z). Answer: This is True. Subset is transitive. (g) Let n 2 N, n > 1 be fixed. (8a, b, c 2 Z)(a ≡ b (mod n) ^ b ≡ c (mod n) ! a ≡ c (mod n)). Answer: This is True. Congruence mod n is transitive. If a ≡ b (mod n), then a − b is a multiple of n. And, if b ≡ c (mod n), then b − c is a multiple of n. Therefore (a − b) + (b − c) = a − c is a multiple of n. (h) (8a, b, c 2 Z)((gcd(a, b) = 1) ^ (gcd(b, c) = 1) ! gcd(a, c) = 1). Answer: This is False. Relatively prime is not transitive. A counterexample is a = 2, b = 3 and c = 4. 5. (Be able to describe the relations determined by Boolean operations.) Describe these relations over the set of Boolean variables. (a) p ^ q = True Answer: This is the set f(1, 1)g. (b) p ^ q = False Answer: This is the set f(0, 0), (0, 1), (1, 0)g. (c) p _ q = True Answer: This is the set f(0, 1), (1, 0), (1, 1)g. (d) p _ q = False Answer: This is the set f(0, 0)g. (e) p ! q = True Answer: This is the set f(0, 0), (0, 1), (1, 1)g. (f) p ! q = False Answer: This is the set f(1, 0)g. (g) p ≡ q = True Answer: This is the set f(0, 0), (1, 1)g. cse 1400 applied discrete mathematics relations and functions 4 (h) p ≡ q = False Answer: This is the set f(0, 1), (1, 0)g. (i) p ⊕ q = True Answer: This is the set f(0, 1), (1, 0)g. (j) p ⊕ q = False Answer: This is the set f(0, 0), (1, 1)g. 6. (Be able to construct all relations in small cases.) List relations in the following instances. (a) The relations from fag to the set of bits B. Answer: i. Æ ii. f(a, 0)g iii. f(a, 1)g iv. f(a, 0), (a, 1)g (b) List all relations on the set of bits B. Answer: i. Æ ix. f(0, 1), (1, 0)g ii. f(0, 0)g x. f(0, 1), (1, 1)g iii. f(0, 1)g xi. f(1, 0), (1, 1)g iv. f(1, 0)g xii. f(0, 0), (0, 1), (1, 0)g v. f(1, 1)g xiii. f(0, 0), (0, 1), (1, 1)g vi. f(0, 0), (0, 1)g xiv. f(0, 0), (1, 0), (1, 1)g vii. f(0, 0), (1, 0)g xv. f(0, 1), (1, 0), (1, 1)g viii. f(0, 0), (1, 1)g xvi. f(0, 0), (0, 1), (1, 0), (1, 1)g 7. (Be able to recognize properties of relations from there representation as ordered pairs.) Which of the relations in problem 6b are reflexive, which are symmetric, which are antisymmetric, and which are transitive? Answer: • Reflexive: 8, 13, 14, 16 • Symmetric: 1, 2, 5, 8, 9, 12, 15, 16 • Antisymmetric: 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14 • Transitive: 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16 ! 8. (Be able to represent a relation as an adjacency matrix.) A relation on an n-element set A = fa0, a1,..., an−1g can be represented by an n × n adjacency matrix of Boolean values. 2 3 a0 a1 ··· an−1 6 7 6 a b b ··· b 7 6 0 0,0 0,1 0,n−1 7 6 7 6 a1 b1,0 b1,1 ··· b1,n−1 7 6 7 6 . 7 4 . ··· . 5 an−1 bn−1,0 bn−1,1 ··· bn−1,n−1 cse 1400 applied discrete mathematics relations and functions 5 Where the entries bi,j 2 B are Boolean values such that bi,j = 1 if ai is related to aj and bi,j = 0 otherwise.
Load more

Recommended publications
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • Mereology Then and Now
    Logic and Logical Philosophy Volume 24 (2015), 409–427 DOI: 10.12775/LLP.2015.024 Rafał Gruszczyński Achille C. Varzi MEREOLOGY THEN AND NOW Abstract. This paper offers a critical reconstruction of the motivations that led to the development of mereology as we know it today, along with a brief description of some questions that define current research in the field. Keywords: mereology; parthood; formal ontology; foundations of mathe- matics 1. Introduction Understood as a general theory of parts and wholes, mereology has a long history that can be traced back to the early days of philosophy. As a formal theory of the part-whole relation or rather, as a theory of the relations of part to whole and of part to part within a whole it is relatively recent and came to us mainly through the writings of Edmund Husserl and Stanisław Leśniewski. The former were part of a larger project aimed at the development of a general framework for formal ontology; the latter were inspired by a desire to provide a nominalistically acceptable alternative to set theory as a foundation for mathematics. (The name itself, ‘mereology’ after the Greek word ‘µρoς’, ‘part’ was coined by Leśniewski [31].) As it turns out, both sorts of motivation failed to quite live up to expectations. Yet mereology survived as a theory in its own right and continued to flourish, often in unexpected ways. Indeed, it is not an exaggeration to say that today mereology is a central and powerful area of research in philosophy and philosophical logic. It may be helpful, therefore, to take stock and reconsider its origins.
    [Show full text]
  • 06. Naive Set Theory I
    Topics 06. Naive Set Theory I. Sets and Paradoxes of the Infinitely Big II. Naive Set Theory I. Sets and Paradoxes of the Infinitely Big III. Cantor and Diagonal Arguments Recall: Paradox of the Even Numbers Claim: There are just as many even natural numbers as natural numbers natural numbers = non-negative whole numbers (0, 1, 2, ..) Proof: { 0 , 1 , 2 , 3 , 4 , ............ , n , ..........} { 0 , 2 , 4 , 6 , 8 , ............ , 2n , ..........} In general: 2 criteria for comparing sizes of sets: (1) Correlation criterion: Can members of one set be paired with members of the other? (2) Subset criterion: Do members of one set belong to the other? Can now say: (a) There are as many even naturals as naturals in the correlation sense. (b) There are less even naturals than naturals in the subset sense. To talk about the infinitely Moral: notion of sets big, just need to be clear makes this clear about what’s meant by size Bolzano (1781-1848) Promoted idea that notion of infinity was fundamentally set-theoretic: To say something is infinite is “God is infinite in knowledge” just to say there is some set means with infinite members “The set of truths known by God has infinitely many members” SO: Are there infinite sets? “a many thought of as a one” -Cantor Bolzano: Claim: The set of truths is infinite. Proof: Let p1 be a truth (ex: “Plato was Greek”) Let p2 be the truth “p1 is a truth”. Let p3 be the truth “p3 is a truth”. In general, let pn be the truth “pn-1 is a truth”, for any natural number n.
    [Show full text]
  • Math 127: Equivalence Relations
    Math 127: Equivalence Relations Mary Radcliffe 1 Equivalence Relations Relations can take many forms in mathematics. In these notes, we focus especially on equivalence relations, but there are many other types of relations (such as order relations) that exist. Definition 1. Let X; Y be sets. A relation R = R(x; y) is a logical formula for which x takes the range of X and y takes the range of Y , sometimes called a relation from X to Y . If R(x; y) is true, we say that x is related to y by R, and we write xRy to indicate that x is related to y by R. Example 1. Let f : X ! Y be a function. We can define a relation R by R(x; y) ≡ (f(x) = y). Example 2. Let X = Y = Z. We can define a relation R by R(a; b) ≡ ajb. There are many other examples at hand, such as ordering on R, multiples in Z, coprimality relationships, etc. The definition we have here is simply that a relation gives some way to connect two elements to each other, that can either be true or false. Of course, that's not a very useful thing, so let's add some conditions to make the relation carry more meaning. For this, we shall focus on relations from X to X, also called relations on X. There are several properties that will be interesting in considering relations: Definition 2. Let X be a set, and let ∼ be a relation on X. • We say that ∼ is reflexive if x ∼ x 8x 2 X.
    [Show full text]
  • Well-Orderings, Ordinals and Well-Founded Relations. an Ancient
    Well-orderings, ordinals and well-founded relations. An ancient principle of arithmetic, that if there is a non-negative integer with some property then there is a least such, is useful in two ways: as a source of proofs, which are then said to be \by induction" and as as a source of definitions, then said to be \by recursion." An early use of induction is in Euclid's proof that every integer > 2 is a product of primes, (where we take \prime" to mean \having no divisors other than itself and 1"): if some number is not, then let n¯ be the least counter-example. If not itself prime, it can be written as a product m1m2 of two strictly smaller numbers each > 2; but then each of those is a product of primes, by the minimality of n¯; putting those two products together expresses n¯ as a product of primes. Contradiction ! An example of definition by recursion: we set 0! = 1; (n + 1)! = n! × (n + 1): A function defined for all non-negative integers is thereby uniquely specified; in detail, we consider an attempt to be a function, defined on a finite initial segment of the non-negative integers, which agrees with the given definition as far as it goes; if some integer is not in the domain of any attempt, there will be a least such; it cannot be 0; if it is n + 1, the recursion equation tells us how to extend an attempt defined at n to one defined at n + 1. So no such failure exists; we check that if f and g are two attempts and both f(n) and g(n) are defined, then f(n) = g(n), by considering the least n where that might fail, and again reaching a contradiction; and so, there being no disagreement between any two attempts, the union of all attempts will be a well-defined function, which is familiar to us as the factorial function.
    [Show full text]
  • Relation Liftings on Preorders and Posets
    Relation Liftings on Preorders and Posets Marta B´ılkov´a1, Alexander Kurz2,, Daniela Petri¸san2,andJiˇr´ı Velebil3, 1 Faculty of Philosophy, Charles University, Prague, Czech Republic [email protected] 2 Department of Computer Science, University of Leicester, United Kingdom {kurz,petrisan}@mcs.le.ac.uk 3 Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic [email protected] Abstract. The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss’s coalgebraic over posets. 1 Introduction Relation lifting [Ba, CKW, HeJ] plays a crucial role in coalgebraic logic, see eg [Mo, Bal, V]. On the one hand, it is used to explain bisimulation: If T : Set −→ Set is a functor, then the largest bisimulation on a coalgebra ξ : X −→ TX is the largest fixed point of the operator (ξ × ξ)−1 ◦ T on relations on X,whereT is the lifting of T to Rel(Set) −→ Rel(Set). (The precise meaning of ‘lifting’ will be given in the Extension Theorem 5.3.) On the other hand, Moss’s coalgebraic logic [Mo] is given by adding to propo- sitional logic a modal operator ∇, the semantics of which is given by applying T to the forcing relation ⊆ X ×L,whereL is the set of formulas: If α ∈ T (L), then x ∇α ⇔ ξ(x) T () α.
    [Show full text]
  • Equivalence Relations ¡ Definition: a Relation on a Set Is Called an Equivalence Relation If It Is Reflexive, Symmetric, and Transitive
    Equivalence Relations ¡ Definition: A relation on a set is called an equivalence relation if it is reflexive, symmetric, and transitive. Recall the definitions: ¢ ¡ ¤ ¤ ¨ ¦ reflexive: £¥¤ §©¨ for all . ¢ ¡ ¤ ¤ ¨ ¨ ¦ ¦ ¦ £ § symmetric: £¥¤ §©¨ when , for . ¢ ¤ ¤ ¨ ¨ ¦ ¦ ¦ £ § transitive: £ §©¨ and £ § implies , ¡ ¤ ¨ for ¦ ¦ . If two elements are related by an equivalence relation, they are said to be equivalent. 1 Examples 1. Let be the relation on the set of English words such that if and only if starts with the same letter as . Then is an equivalence relation. 2. Let be the relation on the set of all human beings such that if and only if was born in the same country as . Then is an equivalence relation. 3. Let be the relation on the set of all human beings such that if and only if owns the same color car as . Then is an not equivalence relation. 2 ¤ Pr is Let ¢ ¢ an oof: is If Since Thus, equi ¤ symmetric. is By refle v ¤ alence definition, be , Congruence for xi ¤ a ¤ v positi e. some relation. £ , ¤ v then e inte £ ¤ § inte ¦ ¤ ger , § , and ger we ¤ 3 . we ha Then Modulo Using v e ha if that v the and , e this,we for ¤ relation only some ¤ ¤ proceed: if inte ger , , so and . Therefore, ¢ ¤ and If ¤ ¤ we congruence ha v e £ ¤ ¤ and , and §"! modulo £ 4 , § and , for is an inte ! is equi , then gers transiti v alence we and v £ ha e. ! v relation e . Thus, § ¦ ¡ Definition: Let be an equivalence relation on a set . The equivalence class of¤ is ¤ ¤ ¨ % ¦ £ § # $ '& ¤ % In words, # $ is the set of all elements that are related to the ¡ ¤ element ¨ .
    [Show full text]
  • On the Necessary Use of Abstract Set Theory
    ADVANCES IN MATHEMATICS 41, 209-280 (1981) On the Necessary Use of Abstract Set Theory HARVEY FRIEDMAN* Department of Mathematics, Ohio State University, Columbus, Ohio 43210 In this paper we present some independence results from the Zermelo-Frankel axioms of set theory with the axiom of choice (ZFC) which differ from earlier such independence results in three major respects. Firstly, these new propositions that are shown to be independent of ZFC (i.e., neither provable nor refutable from ZFC) form mathematically natural assertions about Bore1 functions of several variables from the Hilbert cube I” into the unit interval, or back into the Hilbert cube. As such, they are of a level of abstraction significantly below that of the earlier independence results. Secondly, these propositions are not only independent of ZFC, but also of ZFC together with the axiom of constructibility (V = L). The only earlier examples of intelligible statements independent of ZFC + V= L either express properties of formal systems such as ZFC (e.g., the consistency of ZFC), or assert the existence of very large cardinalities (e.g., inaccessible cardinals). The great bulk of independence results from ZFCLthe ones that involve standard mathematical concepts and constructions-are about sets of limited cardinality (most commonly, that of at most the continuum), and are obtained using the forcing method introduced by Paul J. Cohen (see [2]). It is now known in virtually every such case, that these independence results are eliminated if V= L is added to ZFC. Finally, some of our propositions can be proved in the theory of classes, as formalized by the Morse-Kelley class theory with the axiom of choice for sets (MKC), but not in ZFC.
    [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]
  • Partial Order Relations
    Partial Order Relations We defined three properties of relations: reflexivity, symmetry and transitivity. A fourth property of relations is anti-symmetry. Definition of Antisymmetry:: Let R be a relation on a set A, R is antisymmetric if , and only if, for all a and b in A, if a R b and b R a then a=b. Testing for Antisymmetry of finite Relations: Let R1 and R2 be the relations on {0, 1, 2} defined as follows: a- R1= { (0,2 ), (1, 2), (2, 0)} b- R2 = { (0,0), (0, 1), (0, 2), (1, 1), (1, 2)} Draw a directed graph for R1 and R2 and indicate which relation is antisymmetric? a- R1 is not antisymmetric b- R2 is antisymmetric Testing for Antisymmetry of “Divides” relations. Let R1 be the “divides” relation on the set of all positive integers, and let R2 be the “divides” relation on the set of all integers. + For all a, b ∈ Z , a R1 b ⇔ a | b For all a, b ∈ Z, a R2 b ⇔ a | b a- Is R1 antisymmetric ? Prove or give a counterexample b- Is R2 antisymmetric ? Prove or give a counter example. Solution: -R1 is antisymmetric -R2 is not antisymmetric Partial Order Relations: Let R be a binary relation defined on a set A. R is a partial order relation if, and only if, R is reflexive, antisymmetric and transitive. Two fundamental partial order relations are the “less than or equal to” relation on a set of real numbers and the “subset” relation on a set of sets. The “Subset” Relation: Let A be any collection of sets and define the subset relation ⊆ on A as follows: For all U, V ∈ A U ⊆ V ⇔ for all x, if x ∈ U then x ∈ V.
    [Show full text]
  • Set Theory, Relations, and Functions (II)
    LING 106. Knowledge of Meaning Lecture 2-2 Yimei Xiang Feb 1, 2017 Set theory, relations, and functions (II) Review: set theory – Principle of Extensionality – Special sets: singleton set, empty set – Ways to define a set: list notation, predicate notation, recursive rules – Relations of sets: identity, subset, powerset – Operations on sets: union, intersection, difference, complement – Venn diagram – Applications in natural language semantics: (i) categories denoting sets of individuals: common nouns, predicative ADJ, VPs, IV; (ii) quantificational determiners denote relations of sets 1 Relations 1.1 Ordered pairs and Cartesian products • The elements of a set are not ordered. To describe functions and relations we will need the notion of an ordered pair, written as xa; by, where a is the first element of the pair and b is the second. Compare: If a ‰ b, then ... ta; bu “ tb; au, but xa; by ‰ xb; ay ta; au “ ta; a; au, but xa; ay ‰ xa; a; ay • The Cartesian product of two sets A and B (written as A ˆ B) is the set of ordered pairs which take an element of A as the first member and an element of B as the second member. (1) A ˆ B “ txx; yy | x P A and y P Bu E.g. Let A “ t1; 2u, B “ ta; bu, then A ˆ B “ tx1; ay; x1; by; x2; ay; x2; byu, B ˆ A “ txa; 1y; xa; 2y; xb; 1y; xb; 2yu 1.2 Relations, domain, and range •A relation is a set of ordered pairs. For example: – Relations in math: =, ¡, ‰, ... – Relations in natural languages: the instructor of, the capital city of, ..
    [Show full text]
  • Lecture 2: Well-Orderings January 21, 2009
    Lecture 2: Well-orderings January 21, 2009 3 Well-orderings Definition 3.1. A binary relation r on a set A is a subset of A × A. We also define dom(r) = { x | ∃y.(x, y) ∈ r }, rng(r) = { y | ∃x.(x, y) ∈ r }, and fld(r) = dom(r) ∪ rng(r). Definition 3.2. A binary relation r is a strict partial order iff • ∀x.(x, x) 6∈ r, and • ∀xyz. if (x, y) ∈ r and (y, z) ∈ r then (x, z) ∈ r, that is, if r is irreflexive and transitive. Moreover, iff for all x and y in fld(r), either (x, y) ∈ r or (y, x) ∈ r or x = y, we say r is a strict linear order. Definition 3.3. A strict linear order r is a well-ordering iff every non-empty z ⊆ fld(r) has an r-least member. Remark. For example, the natural numbers N are well-ordered under the normal < relation. However, Z, Q, and Q+ are not. However, we want to be able to talk about well-orderings longer than ω. For example, 0, 2, 4,..., 1, 3, 5,... is an alternative well-ordering of the natural numbers which is longer than ω. Definition 3.4. f : X → X is order-preserving iff for all y, z ∈ X, y < z implies that f(y) < f(z). Theorem 3.5. If hX, <i is a well-ordering and f is order-preserving, then for every y ∈ X, y ≤ f(y). Proof. Suppose otherwise, namely, that there exists some z ∈ X for which f(z) < z.
    [Show full text]