Basic idea of set theory

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Basic idea of set theory

  1. 1. LG511 Basic Ideas of Set Theory, Relations, and Functions Doug Arnold University of Essex doug@essex.ac.uk1 Set Theory: Sets and Membership1.1 SetsA set is a collection of things, (with some clear criterion for membership).Sets are defined by their members — two sets are different if and only if (iff) they have differentmembers. Hence: i. {a, b, c} = {c, b, a} ii. {a, b, c} = {a, a, a, a, a, b, c, c}iii. there is only one empty set, written {}, or ∅iv. Suppose LS and HoD are one and the same, then: {LS , RMA, DJA} = {HoD, RMA, DJA}1.2 Specifying SetsSets can be specified: i. Extensionally: A = {1 , a, Noam Chomsky, lg517 }ii. Intensionally: A = {x : x is greater than 4 } {x |x > 4 } {x : x > 4 } {x : x > 2 + 2 } {y : y <= 5 } etc.1.3 Membership i. a is a member (or element) of {a, b, c}, written: a ∈ {a, b, c}; ii. a is not a member of {b, c, d }, written: a ∈ {b, c, d }iii. The empty set has no membersiv. A singleton set has exactly one member1.4 CardinalitySuppose A = {a, b, c}, then i. |A| = | {a, b, c} | = 3ii. | {x : x > 4 } | = inf 1
  2. 2. 1.5 Sets as IndividualsSets are things — individuals — and so can be members of sets: i. A = {a, {b, c}, lg517 , AR} ii. B = {a, b, c, lg517 , AR}iii. |A| = 4iv. |B| = 5 v. {} = {{}}vi. | {} | = 0vii. | {{}} | = 11.6 Relations between Sets1.6.1 EqualitySets A and B are equal, written A = B, iff they have the same members.Suppose A = {a, b, c, d }, B = {d , c, b, a}, and C = {y : y > 4 } then: i. A = Bii. C = A1.6.2 SubsetSet A is a subset of set B, written A ⊆ B, iff every member of A is a member of B.B is a superset of A, written B ⊇ A, iff A ⊆ B i. {a, b} ⊆ {a, b, c} ii. {a, b, c} ⊇ {a, b}iii. {a, b, c} ⊆ {a, b, c}iv. A ⊆ A v. {} ⊆ ANote: i. a ∈ {a, b, c} ii. a ⊆ {a, b, c}iii. {a} ⊆ {a, b, c}iv. {a} ∈ {a, b, c} v. {a} ∈ {{a} , b, c}1.6.3 Proper SubsetA is a proper (or strict) subset of B, written A ⊂ B, iff A ⊆ B and A = BB is a proper (strict) superset of A, written B ⊃ A, iff B ⊇ A and B = A i. {a, b} ⊂ {a, b, c} ii. {a, b, c} ⊂ {a, b, c}iii. {a, b, c} ⊆ {a, b, c}1.7 Power SetThe power set of set A, written Pow (A) or P(A), is the set consisting of all subsets of A:{S : S ⊂ A}Example: Suppose A = {a, b, c}, then 2
  3. 3.    {}     {a}         {b}        {c}   Pow (A) =  {a, b}     {a, c}         {b, c}        {a, b, c}  Note: i. |Pow (A)| = 2|A| ii. |Pow ({a, b})| = 2|{a,b}| = 22 = 4iii. |Pow ({a, b, c})| = 2|{a,b,c}| = 23 = 81.8 Operations on Sets1.8.1 Venn DiagramsVenn diagrams provide a very convenient and intuitive way of picturing relations between, andoperations on, sets. U B A Figure 1: Venn Diagram1.8.2 ComplementThe complement of set A, written A− or A, is the set of elements not in A:A = {x : x ∈ A} U A Figure 2: A set A and its Complement indicates the set A indicates the Complement of A: (A− or A = U − A) 3
  4. 4. 1.8.3 UnionThe union of sets A and B, written A ∪ B, is the set consisting of all members of A and all membersof B, i.e. every element that is in A or B:A ∪ B = {x : x ∈ A or x ∈ B }Example:{a, b} ∪ {b, c} = {a, b, c} U B A Figure 3: Union of sets A and B indicates A ∪ B, the Union of A and B1.8.4 IntersectionThe intersection of sets A and B, written A ∩ B, is the set consisting of the elements that are in Aand in B, i.e.A ∩ B = {x : x ∈ A and x ∈ B }Example: {a, b, c} ∩ {b, c, d } = {b, c} U B A Figure 4: Intersection of sets A and B indicates A ∩ B, the intersection of A and B1.8.5 DifferenceThe set-difference of sets A and B, written A − B, is the set of elements in A, but not in B:A − B = {x : x ∈ A and x ∈ B }Example: {a, b, c, d } − {c, d } = {a, b}1.9 Properties of OperationsFor a pair of operations ⊕ and ⊗: i. ⊕ is commutative iff A ⊕ B ≡ B ⊕ A ii. ⊕ and ⊗ are associative iff A ⊕ (B ⊗ C) ≡ (A ⊕ B) ⊗ C and A ⊗ (B ⊕ C) ≡ (A ⊗ B) ⊕ Ciii. ⊕ and ⊗ are distributive iff A ⊕ (B ⊗ C) ≡ (A ⊕ B) ⊗ (A ⊕ C) and A ⊗ (B ⊕ C) ≡ (A ⊗ B) ⊕ (A ⊗ C) 4
  5. 5. U B A Figure 5: Difference of sets A and B indicates A − B, the difference of A and BExamples: i. arithmetic +, − (plus, minus) are: Commutative : 2 + 3 = 3 + 2 Associative : 2 + (9 − 5) = (2 + 9) − 5 Non-distributive : 2 + (9 − 5) = (2 + 9) − (2 + 5)ii. set-theoretic ∩ and ∪ are: Commutative (A ∩ B) = (B ∩ A) Associative ((A ∩ B) ∩ C) = (A ∩ (B ∩ A)) Distributive (A ∪ (B ∩ C)) = ((A ∪ B) ∩ (B ∪ C))2 Ordered Pairs and Tuplespair : a, b3-tuple : a, b, cn-tuple : a1 , a2 , a3 , . . . , anCompare: {a, b} = {b, a} but a, b = b, aNote: a1 , a2 = b1 , b2 iff a1 = b1 , and a2 = b22.1 Cartesian ProductA × B = { a, b : a ∈ A and b ∈ B } (the set of ordered pairs a, b st. a is a member of A, and b isa member of B)Example: {a, b, c} × {1 , 2 } =    a, 1 a, 2  b, 1 b, 2 c, 1 c, 2  Note: |A × B| = |A| × |B|3 Relations and Functions3.1 RelationsA relation is a regular association of ‘inputs’ (from a domain) and ‘outputs’ (from a range).i.e. a set of ordered pairs.Example: let L = {a, b, c} and N = {1 , 2 , 3 }, and the relation R ⊂ L × N be as follows: 5
  6. 6.    a, 1  b, 1 c, 2  We could also write either of the following: aR1 R(a, 1) bR1 R(b, 2) cR2 R(c, 2)Note: Relations are defined by their I/Os R1 = R2 just in case they are the same sets of orderedpairs, e.g. R1 : Children × Mothers = R2 : Children × Mothers of siblings3.2 Properties of Relations3.2.1 ReflexivityA relation R isreflexive iff for all x, xRx Examples: is the same height as, ⊆, =irreflexive iff for all x, not xRx Examples: father of, ∈nonreflexive iff neither reflexive nor irreflexive Example: likes3.2.2 SymmetryA relation R is:symmetric iff for all x y, xRy implies yRx Example: is related toasymmetric iff for all x y, xRy implies not yRx Example: mother ofnonsymmetric iff for all x y, xRy implies neither yRx nor ¬yRxantisymmetric iff for all x y, xRy and yRx implies y = x Example: less than or equal to3.2.3 TransitivityA relation R is:transitive iff for all x,y,z xRy and yRz implies xRz Examples: is greater than, is in, is a subset ofintransitive iff for all x,y,z xRy and xRz implies not xRz Example: is immediately to the left ofnontransitive iff neither transitive nor intransitiveEquivalence Relations are reflexive, symmetric, and transitive;Examples: set theoretic equality; is the same age as.3.3 FunctionsA function is a special kind of relation: one which takes (‘maps’) arguments (members of the domain)to values (unique members of the range). 6
  7. 7. (functions are ‘unambiguous’ relations).Recall that a relation is just a set (of ordered pairs) — hence a function is just a special kind of set(of ordered pairs).Examples i. the following function f0 that maps elements of L to elements of N : f0 : L → N :    a, 1  f0 = b, 1 c, 2   f0 (a) = 1 f0 (b) = 1 f0 (c) = 2 ii. f1 : People → Birthdaysiii. f2 : People → Biological FathersThe following relations are not functions: i. { a, 1 , a, 2 , ...} ii. relation between birthdays and peopleiii. relation between fathers and their childreniv. relation between numbers and their square roots3.4 Properties of Functions i. Functions have (not necessarily functional) inverses (the inverse of f is written f −1 ). ii. Functions can be partial, or total.iii. Functions can be onto, into, etc., 1:1, many:1, etc.3.5 Characteristic FunctionsThe Characteristic Function of a set A is a function f : A → {0 , 1 }, such that for all x: i. f (x) = 1 if x ∈ A, andii. f (x) = 0 otherwise.Sets and their characteristic functions are 1:1.4 ReadingPartee et al. (1990, Part A) is a good introduction to set theory.ReferencesBarbara H. Partee, Alice ter Meulen, and Robert E. Wall. Mathematical Methods in Linguistics. Kluwer Academic Publishers, Dordrecht, 1990. 7

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