Math 580—Midterm Review
Definitions—know them
Positive integers N—formal definition
0116_22, 0116_23
Formal definitions of addition, multiplication, and order on N.
0116_25, 0116_27,
Integral domain—formal definition
0114_16.0, 0114_16.1
Absolute value
0123_30.3
Factor
0123_35
Prime integer
0123_37.0
Composite integer
0123_37.0
Common factor
0125_43
Greatest common factor (GCF)
0125_48
Relatively prime integers
0125_52
Congruence modulo m
0125_55
Relation on a set S
0130_58
Equivalence relation on a set S
0130_59
Equivalence classes of an equivalence relation
0130_60
Partition of a nonempty set
0130_62
How to define a partition, given an equivalence relation
0130_61
How to define an equivalence relation, given a partition
0130_63
Zm
0130_65, 0201_65.1
Addition and multiplication on Zm
0201_66, 0201_68
Results—know how to prove
In the integers the additive identity is unique.
0111_11
The additive inverse of an integer is unique.
0111_12
For every a in Z, 0 + a = a.
0111_13
For every a in Z, (-a) + a = 0.
0111_13
For every a, b, c in Z , (b + c)a = ba + ca.
0111_13
For every a in Z, a ⋅ 0 = 0.
0114_13.5
The additive inverse of -a is a.
0114_14.0
For all a, b in Z, a(-b) = -(ab), (-a)b = -(ab),
and (-a)(-b) = ab.
0114_14.5
Addition on N is associative.
0116_25, 0116_26
For every c in N, 1 ≤ c.
0116_28.0
Division algorithm for Z.
Theorem on 0123_31, 0123_32, 0123_33, 0123_34
Congruence mod m is an equivalence relation
0130_57
If Sa and Sb are the equivalence classes containing a, b then
Sa = Sb if and only if aRb, where R is the underlying
equivalence relation.
0130_60
Addition and multiplication on Zm are well-defined.
0201_67, 0201_68
If m > 1 and m | ab and (m, a) = 1, then m | b.
0201_72
If (a, m ) = 1 and (b, m) = 1, then (ab, m) = 1.
0201_74
Results—know the results, but proofs will not be asked on the midterm
General elementary results about integers
0109_6, 0109_7, 0111_8
The integers Z form an integral domain
0111_9, 0111_10.0
In an integral domain, the cancellation law of multiplication holds:
if ab = ac and a ≠ 0, the b = c.
0114_15
The Well-ordering principle for N.
0118_29, 0118_30.0
If m is a positive factor of a nonzero integer n, then m ≤ n.
0123_36
Fundamental Theorem of Arithmetic (unique factorization of integers)
0123_38, 0123_39, 0125_40.0, 0125_40.1, 0125_41, 0125_42
Theorem on Greatest Common Factors
0125_44, 0125_45, 0125_46, 0125_47
How to calculate GCFs and how to find m, n such that (a, b) =
ma + nb.
0125_49, 0125_50, 0125_51
Theorem: p is a prime integer if and only if for all a, b, p divides
ab implies p divides a or p divides b.
0125_53
The equivalence classes of an equivalence relation on a set S
form a partition of S
0130_61
There are infinitely many prime integers
0201_71
If m > 1 and (b, m) = 1, then there exists an integer a such that
ab ≡ 1 (mod m)
0201_73
If (b, m) = 1 and bx ≡ by (mod m), then x ≡ y (mod m)
0301_74
Fermat’s Little Theorem: Let φ(m) be the number of integers
among {1, 2, … m – 1} which are relatively prime to m. Then if
φ(m)
(a, m) = 1, a
≡ 1 (mod m).
0201_75, 0201_76, 0201_77
Notation
∀ (for all), ∃ (there exists), | (divides), ∈ (is an element of),
⇒ (implies)
0118_30.1
a ≡ b (mod m)
0125_56
aRb (used in connection with relations)
0130_58
General background that you need to know
Sets
Functions
Logic
Properties of equality
How to write addition and multiplication tables
0114_17.0, 0114_17.1