LING 106. Knowledge of Meaning
Yimei Xiang
Lecture 2-2
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
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1.1
Relations
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. Jthe capital city ofK “ txUSA, Washingtony, xChina, Beijingy, xFrance, Parisy, ...u
“ txx, yy : y is the capital city of xu
b. JinvitedK “ txAndy, Billyy, xCindy, Dannyy, xEmily, Floriy, ...u
“ txx, yy : x invited yu
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• 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. Jthe capital city ofK Ď tx : x is a country u ˆ ty : y is a cityu
b. Jthe mother ofK Ď tx : x is a human u ˆ ty : y is a humanu
• We can use a mapping diagram to illustrate a relation:
a
1
b
2
c
3
d
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• 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)
1.3
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.
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.
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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.
b.
c.
d.
=
ď
‰
Ď
reflective, symmetric, transitve
(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.
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Functions
• A relation is a function iff each element in the domain is paired with just one element in the range.
(8) f pxq “ x2 for x P N
In particular,
– a function f is a from A (in)to B (written as ‘f : A Ñ B’) iff Dompf q “ A and Rangepf q Ď B.
– a function f is a from A onto B iff Dompf q “ A and Rangepf q “ 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
»
fi
aÑb
b. F “ – c Ñ b fl
dÑe
c. F is a function f with domain ta, b, cu such that f paq “ f pcq “ b and f pdq “ e.
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Exercise: Define conjunctive and, disjunctive or, and negation it is not the case that as functions.
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Characteristic function
• Recall that the following categories can be interpreted as sets of individuals: common nouns, predicative adjectives, intransitive verbs, VPs, ...
...
Annie
(11)
Betty
Cindy
Danny
a. Jwears a hatK “ tAndy, Bettyu
b. Jwears a hatK “ tx : x wears a hatu
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. Jwears a hatK “ txAndy, 1y, xBetty, 1y, xCindy, 0y, xDanny, 0yu
»
fi
Andy Ñ 1
— Betty Ñ 1 ffi
ffi
b. Jwears a hatK “ —
– Cindy Ñ 0 fl
Danny Ñ 0
c. Jwears a hatK “ the function f from individual entities to truth values such that
f pxq “ 1 if x wears a hat, and f pxq “ 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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