of all commitments is maintained
throughout the test. 10. Suppose that in a
three-round obligato game, the teacher
first gives the student the proposition p
q, then the proposition (p r) q, and
finally the proposition q. For which of the
eight possible s

Assuming the truth of the theorem that
states that n is irrational whenever n is a
positive integer that is not a perfect
square, prove that 2 + 3 is irrational.
Computer Projects Write programs with
the specified input and output. 1. Given
the truth valu

proposition involving the propositional
variables p, q, r, and s that is true when
exactly three of these propositional
variables are true and is false otherwise.
8. Show that these statements are
inconsistent: If Sergei takes the job offer
then he will g

computational program or programs you
have written to do these exercises. 1. Look
for positive integers that are not the sum
of the cubes of nine different positive
integers. 2. Look for positive integers
greater than 79 that are not the sum of
the fourth

B B. 33. Find A2 if a) A = cfw_0, 1, 3. b) A =
cfw_1, 2, a, b. 34. Find A3 if a) A = cfw_a. b) A =
cfw_0, a. 35. How many different elements
does A B have if A has m elements and B
has n elements? 36. How many different
elements does A B C have if A has m

implies q): the proposition if p, then q,
which is false if and only if p is true and q
is false converse of p q: the conditional
statement q p contrapositive of pq:
the conditional statement q p inverse
of p q: the conditional statement p
q p q (bicondi

= y z(z = x) (z = y) such that this
statement is true. P1: 1/1 P2: 1/2 QC: 1/1
T1: 2 CH01-7T Rosen-2311T MHIA017Rosen-v5.cls May 13, 2011 15:27
Supplementary Exercises 113 23. Find a
domain for the quantifiers in xy(x = y
z(z = x) (z = y) such that this

bed; he considered these times his most
productive hours for thinking. Descartes
left school in 1612, moving to Paris,
where he spent 2 years studying
mathematics. He earned a law degree in
1616 from the University of Poitiers. At 18
Descartes became disg

usually used to group together elements
with common properties, there is nothing
that prevents a set from having seemingly
unrelated elements. For instance, cfw_a, 2,
Fred, New Jersey is the set containing the
four elements a, 2, Fred, and New Jersey.
So

from consideration.] 49. a) Draw each of
the five different tetrominoes, where a
tetromino is a polyomino consisting of
four squares. b) For each of the five
different tetrominoes, prove or disprove
that you can tile a standard checkerboard
using these te

which he is best known. He also made
fundamental contributions to philosophy.
In 1649 Descartes was invited by Queen
Christina to visit her court in Sweden to
tutor her in philosophy. Although he was
reluctant to live in what he called the
land of bears a

meant by a direct proof, a proof by
contraposition, and a proof by
contradiction of a conditional statement p
q. b) Give a direct proof, a proof by
contraposition and a proof by
contradiction of the statement: If n is
even, then n + 4 is even. 11. a) Des

the final one conclusion: the final
statement in an argument or argument
form valid argument form: a sequence of
compound propositions involving
propositional variables where the truth of
all the premises implies the truth of the
conclusion valid argument

= 0 uniqueness proof: a proof that there is
exactly one element satisfying a specified
property P1: 1/1 P2: 1/2 QC: 1/1 T1: 2
CH01-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 15:27 Supplementary
Exercises 111 RESULTS The logical
equivalences given in

ordered triples (a, b, c), where a A, b
B, and c C. Hence, A B C = cfw_(0, 1, 0),
(0, 1, 1), (0, 1, 2), (0, 2, 0), (0, 2, 1), (0, 2,
2), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 2, 0), (1,
2, 1), (1, 2, 2). Remark: Note that
when A, B, and C are sets, (A B)

(3, 3). We will study relations and their
properties at length in Chapter 9. Using
Set Notation with Quantifiers Sometimes
we restrict the domain of a quantified
statement explicitly by making use of a
particular notation. For example, xS(P
(x) denotes th

show that each set is a subset of the other.
In other words, we can show that if A and
B are sets with A B and B A, then A =
B. That is, A = B if and only if x(x A x
B) and x(x B x A) or
equivalently if and only if x(x A x
B), which is what it means for

codomain, the range, and the assignment
of values to elements of the domain.
EXAMPLE 1 What are the domain,
codomain, and range of the function that
assigns grades to students described in
the first paragraph of the introduction of
this section? Solution:

normals (also known as spies). Knights
always tell the truth, knaves always lie,
and normals sometimes lie and
sometimes tell the truth. Detectives
questioned three inhabitants of the island
Amy, Brenda, and Claireas part of the
investigation of a crime.

integer. Find the truth values of 100
i=1(pi pi+1) and 100 i=1(pi pi+1).
19. Model 16 16 Sudoku puzzles (with
4 4 blocks) as satisfiability problems.
20. LetP (x) be the statement Student x
knows calculus and let Q(y) be the
statement Class y contains a s

to have at least two subsets, the empty set
and the set S itself, that is, S and S
S. THEOREM 1 For every set S, (i) S
and (ii) S S. Proof: We will prove (i) and
leave the proof of (ii) as an exercise. Let S
be a set. To show that S, we must
show that x(

that proceeds by showing that q must be
true when p is true proof by
contraposition: a proof that p q is true
that proceeds by showing that p must be
false when q is false proof by
contradiction: a proof that p is true based
on the truth of the conditiona

using quantifiers, without using the
uniqueness quantifier. 36. Describe a rule
of inference that can be used to prove that
there are exactly two elements x and y in
a domain such that P (x) and P (y) are
true. Express this rule of inference as a
statemen

d) cfw_,cfw_,cfw_,cfw_ 21. Find the power
set of each of these sets, where a and b are
distinct elements. a) cfw_a b) cfw_a, b c) cfw_,
cfw_ 22. Can you conclude that A = B if A
and B are two sets with the same power
set? 23. How many elements does each o

are used to represent the computational
complexity of algorithms, to study the size
of sets, to count objects, and in a myriad
of other ways. Useful structures such as
sequences and strings are special types of
functions. In this chapter, we will
introduc

3 b) c) cfw_ d) cfw_,cfw_ P1: 1 CH02-7T
Rosen-2311T MHIA017-Rosen-v5.cls May
13, 2011 10:24 138 2 / Basic Structures:
Sets, Functions, Sequences, Sums, and
Matrices 60. How many elements does the
successor of a set with n elements have?
Sometimes the numb

first is a subset of the second, the second
is a subset of the first, or neither is a
subset of the other. a) the set of airline
flights from NewYork to New Delhi, the
set of nonstop airline flights from New
York to New Delhi b) the set of people
who spea

compound propositions are logically
equivalent. c) Show in at least two
different ways that the compound
propositions p (r q) and p q
r are equivalent. 5. (Depends on the
Exercise Set in Section 1.3) a) Given a
truth table, explain how to use disjunctive

difference of P and Q are denoted by P
Q, P Q, and P Q, respectively (where
these operations should not be confused
with the analogous operations for sets).
The sum of P and Q is denoted by P + Q.
61. Let A and B be the multisets cfw_3 a, 2
b, 1 c and c

The set of positive integers is infinite.
We will extend the notion of cardinality to
infinite sets in Section 2.5, a challenging
topic full of surprising results. Power Sets
Many problems involve testing all
combinations of elements of a set to see if
t

original checkerboard from 1 to 16,
starting in the first row, moving right in
this row, then starting in the leftmost
square in the second row and moving
right, and so on. Remove squares 1 and
16. To begin the proof, note that square 2
is covered either

[Po62], [Po71], and [Po90]. 14. Describe a
few problems and results about tilings
with polyominoes, as described in [Go94]
and [Ma91], for example. P1: 1 CH02-7T
Rosen-2311T MHIA017-Rosen-v5.cls May
13, 2011 10:24 2 CHAPTER Basic
Structures: Sets, Functio

puzzles without the use of a computer. 5.
Describe the basic rules of WFFN PROOF,
The Game of Modern Logic, developed by
Layman Allen. Give examples of some of
the games included in WFFN PROOF. 6.
Read some of the writings of Lewis
Carroll on symbolic log

logic where n is defined in Exercise 26.
a) 0xP (x) b) 1xP (x) c) 2xP (x) d)
3xP (x) 28. Let P (x, y) be a propositional
function. Show that x y P (x, y) y x
P (x, y) is a tautology. 29. Let P (x) and
Q(x) be propositional functions. Show
that x (P (x) Q(

values to its variables that makes it true
logically equivalent compound
propositions: compound propositions
that always have the same truth values
predicate: part of a sentence that
attributes a property to the subject
propositional function: a statement