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Mathematical Methods in Linguistics

Barbara H. Partee, Alice Ter Meulen, Robert E. Wall (auth.)

Chapter 1

BASIC CONCEPTS OF SET THEORY - all with Video Answers

Educators

Chapter Questions

01:18

Problem 1

Given the following sets:
$$
\begin{array}{ll}
A=\{a, b, c, 2,3,4\} & E=\{a, b,\{c\}\} \\
B=\{a, b\} & F=\emptyset \\
C=\{c, 2\} & G=\{\{a, b\},\{c, 2\}\} \\
D=\{b, c\} &
\end{array}
$$
classify each of the following statements as true or false
(a) $c \in A$
(g) $D \subset A$ (m) $B \subseteq G$
(b) $c \in F$
(h) $A \subseteq C$
(n) $\{B\} \subseteq G$
(c) $c \in E$
(i) $D \subseteq E$
(o) $D \subseteq G$
(d) $\{c\} \in E$
(j) $F \subseteq A$
(p) $\{D\} \subseteq G$
(e) $\{c\} \in C$
(k) $E \subseteq F$
(q) $G \subseteq \bar{A}$
(f) $B \subseteq A$
(1) $B \in G$
(r) $\{\{c\}\} \subseteq E$

Amy Jiang
Amy Jiang
Numerade Educator
01:07

Problem 2

For any arbitrary set $S$,
(a) is $S$ a member of $\{S\}$ ?
(b) is $\{S\}$ a member of $\{S\}$ ?
(c) is $\{S\}$ a subset of $\{S\}$ ?
(d) what is the set whose only member is $\{S\}$ ?

Tanishq Gupta
Tanishq Gupta
Numerade Educator

Problem 3

Write a specification by rules and one by predicates for each of the following sets. Remember that there is no order assumed in the list, so you cannot use words like 'the first' or 'the latter'. Recall also that a recursive rule may contain more than one if-then statement.
(a) $\{5,10,15,20, \ldots\}$
(b) $\{7,17,27,37, \ldots\}$
(c) $\{300,301,302, \ldots, 399,400\}$
(d) $\{3,4,7,8,11,12,15,16,19,20, \ldots\}$
(e) $\{0,2,-2,4,-4,6,-6, \ldots\}$
(f) $\{1,1 / 2,1 / 4,1 / 8,1 / 16, \ldots\}$

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02:33

Problem 4

Consider the following sets:
$$
\begin{array}{ll}
S 1=\{\{0\},\{A\}, A\} & S 6=0 \\
S 2=A & S 7=\{0\} \\
S 3=\{A\} & S 8=\{\{0\}\} \\
S 4=\{\{A\}\} & S 9=\{0,\{0\}\} \\
S 5=\{\{A\}, A\} &
\end{array}
$$
Answer the following questions Remember that the members of a set are the items separated by commas, if there is more than one, between the outermost braces only; a subset is formed by enclosing within braces zero or more of the members of a given set, separated by commas.
(a) Of the sets $S 1 \cdot S 9$ which are members of $S 1$ ?
(b) which are subsets of $S 1$ ?
(c) which are members of $S 9$ ?
(d) which are subsets of $S 9$ ?
(e) which are members of $S 4$ ?
(f) which are subsets of $S 4$ ?

Aman Gupta
Aman Gupta
Numerade Educator
00:13

Problem 5

Specify each of the following sets by listing its members:
(a) $\wp\{a, b, c\}$
(d) $\varphi\{\emptyset\}$
(b) $\varphi\{a\}$
(e) $\rho p\{a, b\}$
(c) $\varphi \emptyset$

Amy Jiang
Amy Jiang
Numerade Educator
01:23

Problem 6

Given the sets $A, \ldots, G$ as in Exercise 1, list the members of each of the following:
(a) $B \cup C$
(g) $A \cap E$
(m) $B-A$
(b) $A \cup B$
(h) $C \cap D$
(n) $C-D$
(c) $D \cup E$
(i) $B \cap F$
(o) $E-F$
(d) $B \cup G$
(j) $C \cap E$
(p) $F-A$
(e) $D \cup F$
(k) $B \cap G$
(q) $G-B$
(f) $A \cap B$
(1) $A-B$

Aman Gupta
Aman Gupta
Numerade Educator

Problem 7

Given the sets in Exercise 1, assume that the universe of discourse is $\bigcup\{A, B, C, D, E, F, G\}$. List the members of the following sets:
(a) $(A \cap B) \cup C$
(h) $D^{\prime} \cap E^{\prime}$
(b) $A \cap(B \cup C)$
(i) $F \cap(A-B)$
(c) $(B \cup C)-(C \cup D)$
(j) $(A \cap B) \cup U$
(d) $A \cap(C-D)$
(k) $(C \cup D) \cap U$
(e) $(A \cap C)-(A \cap D)$
(l) $C \cap D^{\prime}$
(f) $G^{\prime}$
(m) $G \cup F^{\prime}$
(g) $(D \cup E)^{\prime}$
(n) $(B \cap C)^{\prime}$

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07:28

Problem 8

Let $A=\{a, b, c\}, B=\{c, d\}$ and $C=\{d, e, f\}$.
(a) What are:
(i) $A \cup B$
(v) $B \cup \emptyset$
(ii) $A \cap B$
(vi) $A \cap(B \cap C)$
(iii) $A \cup(B \cap C)$
(vii) $A-B$
(iv) $C \cup A$
(b) Is a member of $\{A, B\}$ ?
(c) Is $a$ a member of $A \cup B$ ?

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:04

Problem 9

Show by using the set-theoretic equalities in Figure 1-7 for any sets $A$, $B$, and $C$,
(a) $\left((A \cup C) \cap\left(B \cup C^{\prime}\right)\right) \subseteq(A \cup B)$
(b) $A \cap(B-A)=0$

Doruk Isik
Doruk Isik
Numerade Educator
02:58

Problem 10

Show that the Distributive Law $4(a)$ is true by constructing Venn diagrams for $X \cup(Y \cap Z)$ and $(X \cup Y) \cap(X \cup Z)$.

Harshita Goel
Harshita Goel
Numerade Educator
00:56

Problem 12

The symmetric difference of two sets $A$ and $B$, denoted $A+B$, is defined as the set whose members are in $A$ or in $B$ but not in both $A$ and $B$, i.e.
$$
A+B={ }_{\operatorname{def}}(A \cup B)-(A \cap B)
$$
(a) Draw the Venn diagram for the symmetric difference of two sets.
(b) Show that $A+B=(A-B) \cup(B-A)$ by means of the settheoretic equalities in Figure 1-7. Verify that the Venn diagram for $(A-B) \cup(B-A)$ is equivalent to that in (a).
(c) Show that for all sets $A$ and $B, A+B=B+A$.
(d) Express each of the following in terms of union, intersection, and complementation, and simplify using the set-theoretic equalities.
(i) $A+A$
(iv) $A+B$, where $A \subseteq B$
(ii) $A+U$
(v) $A+B$, where $A \cap B=0$
(iii) $A+\emptyset$
(e) Show that $((A-B)+(B-A))=A+B$
(f) Show that $(A+B) \subseteq B$ iff $A \subseteq B$

Doruk Isik
Doruk Isik
Numerade Educator

Problem 12

Call adjectives which are correctly predicated of themselves 'autological' and those which are not, 'heterological.' For example, 'English' and 'short' are autological, but 'French' and 'long' are heterologial, Show that when we ask whether the adjective 'heterological' is heterological or autological we are led to a contradiction like that in Russell's Paradox. This is known as Grelling's Paradox.

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