Set Theory, Relations, and Functions (I)

LING 106. Knowledge of Meaning Lecture 2-1 Yimei Xiang Jan 30, 2016

Set theory, relations, and functions (I)

1 Set theory

1.1 What is a set? •A set is an abstract collection of distinct objects. Those objects are the members/elements of this set. In set theory, the concept ‘set’ is extensional (as oppose to intensional), namely, we don’t bother about the ways in which the members of a set are chosen.

Exercise: Which of the following expressions correctly describe the features of set members?

a. Distinct to each other b. Ordered c. Relevant to each other d. The overall quantity is ﬁnite

• Some special sets

– Singleton set: A set with exactly one member E.g. tau, tta, buu – Empty/null set H: the set that has no member

• Convention of set-theoretic notations

(1) a. (Ordinary) sets: A, B, C, ... b. Members of a set: a, b, c, ..., x, y, z c. Sets of sets: A , B, C d. Cardinality of A: |A| or #(A) e. membership relation: P

Exercise: What is the cardinality of the set ta, a, b, b, cu?

Exercise: Is tHu a singleton set or an empty set? What about tH, Hu?

1 • Ways to deﬁne a set

– list/roster notation: listing all its members

(2) E = {4, 6, 8, ... }

Note: The order of the elements doesn’t matter. Elements can be listed multiple times. – predicate/abstraction notation: stating a condition that is satisﬁed by all and only the elements of the set to be deﬁned Schema: tx | P pxqu, where x is a variable, and P is a predicate.

(3) a. {x | x is an even number greater than 3} Read as: “the set of all x such that x is an even number greater than 3” b. {x | x is evenly divisible by 2 and is greater than or equal to 4}

– recursive rules: using rules which generates/deﬁnes the members of this set.

(4) a. 4 P E (the basis of recursion) b. If x P E, then x ` 2 P E (to generate new elements from the elements deﬁned before) c. Nothing else belongs to E. (to restrict the deﬁned set to the elements generated by rules a and b)

Exercise: Identify the set of positive integers in three different ways.

2 1.2 Set-theoretic relations and operations • Relations of sets

(5) a. Identity/Equality Two sets are identical/equal iff they have exactly the same members. A = B iff for every x: x P A iff x P B. b. Subsets i. A is a subset of B iff every element of A is also an element of B. A Ď B iff for every x: if x P A, then x P B. ii. A a proper subset of B iff A is a subset of B but not identical to B. A Ă B iff A Ď B and A ‰ B. c. Power sets i. Given a set A, the set of all the subsets of A is called the power set of A, written as P(A). ii. ∅ is a subset of every set.

Exercise: Let S be an arbitrary set, consider:

a. is S a member of tSu? b. is tSu a member of tSu? c. is tSu a subset of tSu? d. what is the set whose only member is tSu?

3 • Operations on sets

(6) a. Union A Y B “ tx | x P A or x P Bu (or equivalently: tA, Bu) b. Intersection Ť A X B “ tx | x P A and x P Bu (or equivalently: tA, Bu) c. Difference Ş A ´ B “ tx | x P A and x R Bu also called: the (relative) complement of B in A d. (Absolute) Complement A1 “ U ´ A “ tx | x P U and x R Au

We can use Venn diagrams to illustrate these operations.

A A B A B A B U

Union A Y B Intersection A X B Difference A ´ B Complement A1

Exercise: Simplify the following set-theoretic notations

(7) a. tx : x “ au b. tx | x ‰ xu c. pA ´ Bq ´ B1

Exercise: Use Venn diagrams to verify DeMorgan’s Law.

(8) DeMorgan’s Law (in set theory) a. pA Y Bq1 “ A1 X B1 b. pA X Bq1 “ A1 Y B1

4 1.3 Applications of set theory in natural language semantics • The following categories denote sets of individuals:

– common noun (ﬂower, cat) – predictive adjective (happy, gray) – VP and intransitive verb (wears a hat, run, meow)

For example:

(9) a. cat “ tx | x is a catu J K b. wears a hat “ tx | x wears a hatu J K • Quantiﬁcational determiners (e.g., every, some) denote relations of sets

(10) a. Every cat meows. tx | x is a catu Ď tx | x meowsu b. Some cat meows. tx | x is a catu X tx | x meowsu ‰ H

Exercise: Use set-theoretic notations to represent the meaning of the following sentences.

(11) a. Every gray cat meows. b. No cat barks. c. Two cats are meowing. d. Most cats meow.