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: =, ą, ‰, ... – 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. ď 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. {x2, 3y, x5, 4y, x0, 3y, x4, 1y} b. {x3, ´1y, x2, ´2y, x0, 2y, x2, 1y}

• 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.

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