Math 3000 Running Glossary

Last Updated on: April 9, 2010
Chapter 1:
1. A set:
A collection of objects called elements.
2. The empty set (∅):
A set containing no elements.
3. Natural Numbers (N):
{1, 2, 3, . . .}
Math 3000
Running Glossary
4. Integers (Z):
{. . . , −3, −2, −1, 0, 1, 2, 3, . . .} or {x : x ∈ N} ∪ {−x : x ∈ N} ∪ {0}
5. Rational Numbers (Q):
m
: m ∈ Z and n ∈ Z\{0} or
n
m
n
: m ∈ Z and n ∈ Z where n = 0
6. Real Numbers (R):
Any number that can be written in decimal form. (note that this is not the precise definition, but
it is informally correct)
7. Complex Numbers (C):
{a + bi : a ∈ R and b ∈ R}. Note that it is understood there that i = √ −1.
8. Cardinality of a set S (denoted |S|):
The number of elements in S.
9. A ⊆ B:
A is a subset of B if every element of A is also an element of B.
10. Open Interval (a, b) :
(a, b) = {x ∈ R : a < x < b}
11. Closed Interval [a, b]:
[a, b] = {xinR : a ≤ x ≤ b}
12. Half-open of half-closed intervals [a, b) or (a, b]:
[a, b) = {x ∈ R : a ≤ x < b}
(a, b] = {x ∈ R : a < x ≤ b}
13. Power Set of a set A (P(A)):
P(A) = {S : S is a subset of A}
(Note that the empty set and the set A are both members of the power set)
14. Union A ∪ B:
A ∪ B = {x : x ∈ A or x ∈ B}
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15. Intersection A ∩ B:
A ∩ B = {x : x ∈ A and x ∈ B}
16. Disjoint Sets:
If two sets A and B have no elements in common, then A ∩ B = ∅ and A and B are said to be
disjoint.
17. Set Different A − B or A\B:
A − B = A\B = {x : x ∈ A and x /∈ B}
18. Complement Ā:
Ā = {x : x ∈ U and x /∈ A} (U is the universal set)
19. Indexed Union:
Sα = {x : x ∈ Sα for some α ∈ I}
α∈I
20. Indexed Intersection:
Sα = {x : x ∈ Sα for all α ∈ I}
α∈I
21. Partition of a set:
A partition of a set A is a collection S of nonempty subsets of A such that every element of A
belongs to exactly one subset of S.
An alternate definition: A partition of a set A is a collection S of subsets of A such that
(1) X = ∅ for every X ∈ S
(2) for every two sets X, Y ∈ S, either X = Y of X ∩ Y = ∅
(3) ∪X∈SX = A.
22. Cartesian Product:
A × B = {(a, b) : a ∈ A and b ∈ B}
Chapter 2:
23. Statement:
A statement is a declarative sentence or assertion that is true or false (but not both).
24. Open Sentence:
An open sentence is a declarative sentence that contains one or more variables.
25. Disjunction:
The statement P or Q. Also denoted P ∨ Q
26. Conjunction:
The statement P and Q. Also denoted P ∧ Q
27. Conditional Statement:
The statement: If P then Q. Also denoted P =⇒ Q.
28. Converse:
The converse of P =⇒ Q is Q =⇒ P
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29. The Biconditional:
The statement: P =⇒ Q along with its converse Q =⇒ P . Another notation: (P ⇐⇒ Q).
Another notation: (P if and only if Q).
30. Tautology:
A statement is a tautology if it is true for all possible combinations of truth values.
31. Logical Equivalence:
When two statements have the same truth values for all combinations of truth values.
32. The ∀ quantifier:
for all
33. The ∃ quantifier:
there exists
Chapter 3:
34. Axiom:
A true mathematical statement whose truth is accepted without proof.
35. Theorem:
A true mathematical statement whose truth can be verified.
36. Corollary:
A mathematical result that can be deduced from some earlier result.
37. Lemma:
A mathematical result that is useful in establishing the truth of some other result.
38. Trivial Proof:
If Q(x) is true for all x ∈ S (regardless of the truth of P (x)), then P (x) =⇒ Q(x) is true trivially.
39. Vacuous Proof:
If P (x) is false for all x ∈ S (regardless of the truth of Q(x)), then P (x) =⇒ Q(x) is true
vacuously.
40. Direct Proof:
To prove P (x) =⇒ Q(x) with a direct proof, assume that P (x) is true and prove that Q(x) is
true.
41. An even integer:
An integer n is even if there exists an integer k such that n = 2k.
42. A odd integer:
An integer n is odd if there exists an integer k such that n = 2k + 1.
43. Contrapositive:
The contrapositive of P =⇒ Q is ∼ Q =⇒ ∼ P .
44. Without Loss of Generality (WOLOG):
This phrase indicates that the proofs of two situations are similar, so the proof of only one of these
is needed.
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Chapter 4:
45. a divides b. Denoted: a|b
We say that a divides b if there is an integer c such that b = ac.
46. b is a multiple of a:
If a|b the we say that b is a multiple of a.
47. a is a divisor of b:
If a|b then we say that a is a divisor of b.
48. a does not divide b. Denoted (a |b):
There is not an integer c such that b = ac.
49. a is congruent to b modulo n. Denoted a ≡ b (mod n):
a ≡ b (mod n) if n|(a − b). Think of this as: a and b have the same remainder when divided by n.
50. Absolute Value:
51. Triangle Inequality:
|x + y| ≤ |x| + |y|
|x| =
x if x ≥ 0
−x if x < 0.
52. Relative Complement:
The set A − B is called the relative complement of B in A. The relative complement of A in U is
called the complement of A and is denoted Ā
53. Fundamental Properties of Set Operations (this is technically a theorem)
(a) Commutative Laws
i. A ∪ B = B ∪ A
ii. A ∩ B = B ∩ A
(b) Associative Laws
i. A ∪ (B ∪ C) = (A ∪ B) ∪ C
ii. A ∩ (B ∩ C) = (A ∩ B) ∩ C
(c) Distributive Laws
i. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
ii. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(d) DeMorgan’s Laws
i. A ∪ B = A ∩ B
ii. A ∩ B = A ∪ B
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Chapter 5
(There aren’t many new definitions in chapter 5, but some very powerful proof techniques are
introduced.)
54. Counterexample:
An element y ∈ S is called a counterexample of the statement (∀x ∈ S, R(x)) if R(y) is false.
Chapter 6
(We only cover sections 6.1 and 6.2)
55. Well Ordered Set
A set, S, is called well ordered if every nonempty subset of S has a least element.
(An example is the natural numbers (see below). An example of a set that is not well ordered is
the set of complex numbers.)
56. Well Ordering Priciple
The set of natural numbers (N) is well ordered. (This is really an axiom)
57. The Principle of Mathematical Induction (theorem 6.1):
For each positive integer n, let P (n) be a statement. If
(1) P (1) is true and
(2) the implication
is true for every positive integer k,
then P (n) is true for every positive integer n.
If P (k), then P (k + 1)
58. The Second Principle of Mathematical Induction (theorem 6.8):
For a fixed integer m, let S = {i ∈ Z : i ≥ m}. For each integer n ∈ S, the P (n) be a statement.
If
(1) P (m) is true and
(2) the implication
is true for every k ∈ S
then P (n) is true for every integer n ∈ S.
Chapter 8
(We only cover sections 8.1 - 8.4)
if P (k), then P (k + 1)
59. Relation
A relation, R, from A to B is a subset of A × B. In other words, R is a set of ordered pairs,
where the first coordinate of the pair belongs to A and the second coordinate belongs to B.
Notation: aRb reads ”a is related to b.”
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60. Domain of a relation
dom(R) = {a ∈ A : (a, b) ∈ R for some b ∈ B}
61. Range of a relation
ran(R) = {b ∈ B : (a, b) ∈ R for some a ∈ A}
62. Relation on a set A
A relation on a set A is a relation from A to A. In other words, a relation on set A is a subset of
A × A.
63. Reflexive property
A relation is reflexive if xRx for every x in the set A.
64. Symmetric property
A relation R defined on a set A is symmetric if whenever xRy, then yRx for all x, y ∈ A
65. Transitive property
A relation R defined on a set A is transitive if whenever xRy and yRz, then xRz for all x, y, z ∈ A.
66. Distance
The distance between two real numbers a and b is |a − b|.
67. Equivalence Relation
A relation R on a set A is called an equivalence relation if R is reflexive, symmetric, and
transitive.
68. Equivalence Class
For an equivalence relation R defined on a set A and for a ∈ A, the set
[a] = {x ∈ A : xRa}
consisting of all elements in A that are related to a, is called an equivalence class.
Chapter 9
(We will skip section 9.2 and 9.7)
69. Function
Let A and B be nonempty sets. By a function f from A to B, written f : A → B, we mean
a relation from A to B with the property that every element a of A is the first coordinate of
exactly one ordered pair in f.
70. Domain of a function
If f : A → B is a function, the set A is the domain of f.
71. Codomain of a function
If f : A → B is a function, the set B is the codomain of f.
72. Image
If (a, b) ∈ f, then we write b = f(a) and refer to b as the image of a.
73. Map (mapping)
If f is a function it is sometimes called a map or a mapping.
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74. Range of a function
ran(f) = {f(x) ∈ B : x ∈ A}
75. Graph of a function
G(f) = {(x, f(x)) : x ∈ A}
76. Equal functions
Two functions, f and g, are equal if f(a) = g(a) for every a ∈ A.
77. One-to-One function (Injective Function)
A function f : A → B is one-to-one if whenever f(x) = f(y), where x, y ∈ A, then x = y.
78. Onto function (Surjective Function)
A function f : A → B is called onto if every element of the codomain B is the image of some
element of A.
79. Bijective Function
A function f : A → B is called bijective if it is both one-to-one and onto (injective and surjective).
80. A function is well-defined if
A function f is called well defined if for (a, b), (a, c) ∈ f we have b = c.
81. Composition of Functions
Let f : A → B and g : B → C. The composition g ◦ f is a function (g ◦ f) : A → C defined by
(g ◦ f)(a) = g(f(a)) for all a ∈ A.
82. Inverse Relation
R −1 = {(b, a) : (a, b) ∈ R}
83. Inverse Function theorem (not really a definition, but very important)
Let f : A → B be a function. Then the inverse relation f −1 is a function from B to A if and only
if f is bijective. Furthermore, if f is bijective, then f −1 is also bijective.
84. Property of inverse functions
If f : A → B is a bijective function (so that an inverse exists), then f −1 ◦ f = iA and f ◦ f −1 = iB
85. Indicator Function (used for problem 9.47)
Let U be some universal set and A be a subset of U. The indicator function χA : U → {0, 1} is
defined by
Chapter 10
χA(x) =
1, x ∈ A
0, x ∈ A
(This will be our last chapter. We will cover all that times allows)
86. Finite Set:
A set S is finite if either S = ∅ or |S| = n for some n ∈ N.
87. Infinite Set: (2 definitions)
(a) A set if infinite if it is not finite
(b) A set S is infinite if it contains a proper subset that can be put in one-to-one correspondence
with S.
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88. Sets with the same cardinality
Two sets A and B are said to have the same cardinality, written |A| = |B|, if either A of B are
both empty of there is a bijective function f from A to B.
89. Numerically Equivalent Sets
Two sets having the same cardinality are referred to as numerically equivalent sets.
90. Denumerable Set
A set S is called denumerable if |S| = |N|.
91. Countable Set
A set is countable if it is either finite or denumerable.
92. Uncountable Set
A set is uncountable if it is not countable.
93. Meaning of “smaller cardinality”
A set A is said to have smaller cardinality than a set B, written as a|A| < |B|, if there exists a
one-to-one function from A to B but NO bijective function from A to B.
94. ℵ0 (aleph nought)
The cardinality of the set N of natural numbers is denoted ℵ0.
95. “little c”
The cardinality of the set R or real numbers is denoted c.
96. The Continuum Hypothesis
There exists NO set S such that ℵ0 < |S| < c.
97. Let A and B be sets and let f : A → B. Assume D ⊂ A. The restriction, f1, of f to D is
f1 = {(x, y) ∈ f : x ∈ D}
98. AND THE LAST AMAZING FACT!!
The sets P(N) and R are numerically equivalent! (The power set of the natural numbers is the
same “size” as the real numbers)
99. QED
100.
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