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Set Theory, Relations, and Functions (II)

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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, ... (2) a. the capital city of “ txUSA; Washingtony; xChina; Beijingy; xFrance; Parisy; :::u J K “ txx; yy : y is the capital city of xu b. invited “ txAndy; Billyy; xCindy; Dannyy; xEmily; Floriy; :::u J K “ txx; yy : x invited yu 1 • R is a relation from A to B iff R is a subset of the Cartesian product A ˆ B, written as R Ď A ˆ B. R is a relation in A iff R is a subset of the Cartesian product A ˆ A, written as R Ď A ˆ A. (3) a. the capital city of Ď tx : x is a country u ˆ ty : y is a cityu J K b. the mother of Ď tx : x is a human u ˆ ty : y is a humanu J K • We can use a mapping diagram to illustrate a relation: a 1 b 2 c 3 d 4 • A and B are the domain and range of R respectively, iff A ˆ B is the smallest Cartesian product of which R is a subset. For example: (4) Let R “ tx1; ay; x1; by; x2; by; x3; byu, then: DompRq “ t1; 2; 3u, RangepRq “ ta; bu Discussion: Why is that the following definitions are problematic? Given counterexamples. (5) a. A and B are the domain and range of R iff R “ A ˆ B. b. A and B are the domain and range of R iff R Ď A ˆ B. 1.3 Properties of relations • Properties of relations: reflexivity, symmetry, transitivity Given a set A and a relation R in A, ... – R is reflexive iff for every x in A, xx; xy P R; otherwise R is nonreflexive. If there is no pair of the form xx; xy in R, then R is irreflexive. – R is symmetric iff for every xy in A, if xx; yy P R, then xy; xy P R; otherwise R is nonsymmetric. If there is no pair xx; yy in R such that the pair xy; xy is in R, then R is asymmetric. – R is transitive iff for every xyz in A, if xx; yy P R and xy; zy P R, then xx; zy P R; otherwise R is nontransitive. If there are no pairs xx; yy and xy; zy in R such that the pair xx; zy is in R, then R is intransitive. 2 Exercise: Identify the properties of the following relations w.r.t. reflexivity, symmetry, and transitivity. (Make the strongest possible statement. For example, call a relation ‘irreflexive’ instead of ‘nonreflexive’, if satisfied.) (6) a. = reflective, symmetric, transitve b. ¤ c. ‰ d. Ď (7) a. is a sister of b. is a mother of • It is helpful in assimilating the notions of reflexivity, symmetry and transitivity to represent them in relational diagrams. If x is related to y (namely, xx; yy P R), an arrow connects the corresponding points. 2 Functions • A relation is a function iff each element in the domain is paired with just one element in the range. 2 (8) fpxq “ x for x P N In particular, – a function f is a from A (in)to B (written as ‘f: A Ñ B’) iff Dompfq “ A and Rangepfq Ď B. – a function f is a from A onto B iff Dompfq “ A and Rangepfq “ B. Exercise: For each set of ordered pairs, consider: what is its domain and range? Is it a function? If it is, draw a mapping diagram to illustrate this function. (9) a. fx2; 3y; x5; 4y; x0; 3y; x4; 1yg b. fx3; ´1y; x2; ´2y; x0; 2y; x2; 1yg • We may specify functions with lists, tables, or words. (10) a. F “ txa; by; xc; by; xd; eyu a Ñ b b. F “ c Ñ b » fi d Ñ e c. F is a– functionflf with domain ta; b; cu such that fpaq “ fpcq “ b and fpdq “ e. 3 Exercise: Define conjunctive and, disjunctive or, and negation it is not the case that as functions. 3 Characteristic function • Recall that the following categories can be interpreted as sets of individuals: common nouns, predica- tive adjectives, intransitive verbs, VPs, ... ... Annie Betty Cindy Danny (11) a. wears a hat “ tAndy; Bettyu J K b. wears a hat “ tx : x wears a hatu J K Alternatively, the semantics of these expressions can also be modeled as functions from the set of individual entities to the set of truth values. (12) a. wears a hat “ txAndy; 1y; xBetty; 1y; xCindy; 0y; xDanny; 0yu J K Andy Ñ 1 Betty Ñ 1 b. wears a hat “ » fi J K Cindy Ñ 0 — Danny Ñ 0 ffi — ffi c. wears a hat “ the– function f fromfl individual entities to truth values such that J K fpxq “ 1 if x wears a hat, and fpxq “ 0 otherwise. Characteristic function: a function whose range is the set of possible truth values t1; 0u. Exercise: provide the characteristic function for wears a bowtie. 4.
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