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List of Symbols

Subject Symbol Meaning Page

Logic ∼p not p 25 p ∧ q p and q 25 p ∨ q p or q 25 p ⊕ q or p XOR q p or q but not both p and q 28 P ≡ Q P is logically equivalent to Q 30 p→ q if p then q 40 p q p if and only if q 45 ∴ therefore 51 P(x) predicate in x 97

P(x)⇒ Q(x) every element in the truth set for P(x) is in 104 the truth set for Q(x)

P(x)⇔ Q(x) P(x) and Q(x) have identical truth sets 104 ∀ for all 101 ∃ there exists 103

Applications of Logic NOT NOT-gate 67

AND AND-gate 67

OR OR-gate 67

NAND NAND-gate 75

NOR NOR-gate 75

| Sheffer stroke 74

↓ Peirce arrow 74 n2 number written in binary notation 78

n10 number written in decimal notation 78

n16 number written in hexadecimal notation 91

Number Theory and Applications

d | n d divides n 170 d |/ n d does not divide n 172 n div d the integer quotient of n divided by d 181

n mod d the integer remainder of n divided by d 181

�x� the floor of x 191 �x� the ceiling of x 191 |x | the absolute value of x 187 gcd(a, b) the greatest common divisor of a and b 220

x := e x is assigned the value e 214

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Subject Symbol Meaning Page

Sequences . . . and so forth 227 n∑

k=m ak the summation from k equals m to n of ak 230

n∏ k=m

ak the product from k equals m to n of ak 223

n! n factorial 237 Set a ∈ A a is an element of A 7 Theory a /∈ A a is not an element of A 7

{a1, a2, . . . , an} the set with elements a1, a2, . . . , an 7 {x ∈ D | P(x)} the set of all x in D for which P(x) is true 8 R,R−,R+,Rnonneg the sets of all real numbers, negative real 7, 8

numbers, positive real numbers, and nonnegative real numbers

Z,Z−,Z+,Znonneg the sets of all integers, negative integers, 7, 8 positive integers, and nonnegative integers

Q,Q−,Q+,Qnonneg the sets of all rational numbers, negative 7, 8 rational numbers, positive rational numbers, and nonnegative rational numbers

N the set of natural numbers 8

A ⊆ B A is a subset of B 9 A �⊆ B A is not a subset of B 9 A = B A equals B 339 A ∪ B A union B 341 A ∩ B A intersect B 341 B − A the difference of B minus A 341 Ac the complement of A 341

(x, y) ordered pair 11

(x1, x2, . . . , xn) ordered n-tuple 346

A × B the Cartesian product of A and B 12 A1 × A2 × · · · × An the Cartesian product of A1, A2, . . . , An 347 ∅ the empty set 361 P(A) the power set of A 346

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List of Symbols

Subject Symbol Meaning Page

Counting and N (A) the number of elements in set A 518 Probability P(A) the probability of a set A 518

P(n, r) the number of r -permutations of a set of 553 n elements(n

r

) n choose r , the number of r -combinations 566 of a set of n elements, the number of r -element subsets of a set of n elements

[xi1 , xi2 , . . . , xir ] multiset of size r 584 P(A | B) the probability of A given B 612

Functions f : X → Y f is a function from X to Y 384 f (x) the value of f at x 384

x f→y f sends x to y 384

f (A) the image of A 397

f −1(C) the inverse image of C 397

Ix the identity function on X 387

bx b raised to the power x 405, 406

expb(x) b raised to the power x 405, 406

logb(x) logarithm with base b of x 388

F−1 the inverse function of F 411

f ◦ g the composition of g and f 417 Algorithm x ∼= y x is approximately equal to y 237 Efficiency O( f (x)) big-O of f of x 727

�( f (x)) big-Omega of f of x 727

�( f (x)) big-Theta of f of x 727

Relations x R y x is related to y by R 14

R−1 the inverse relation of R 444

m ≡ n (mod d) m is congruent to n modulo d 473 [a] the equivalence class of a 465 x � y x is related to y by a partial order relation � 502

Continued on first page of back endpapers.

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DISCRETE MATHEMATICS

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DISCRETE MATHEMATICS WITH APPLICATIONS

FOURTH EDITION

SUSANNA S. EPP DePaul University

Australia · Brazil · Japan · Korea ·Mexico · Singapore · Spain · United Kingdom · United States

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Cover Photo: The stones are discrete objects placed one on top of another like a chain of careful reasoning. A person who decides to build such a tower aspires to the heights and enjoys playing with a challenging problem. Choosing the stones takes both a scientific and an aesthetic sense. Getting them to balance requires patient effort and careful thought. And the tower that results is beautiful. A perfect metaphor for discrete mathematics!

DiscreteMathematics with Applications, Fourth Edition Susanna S. Epp

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