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### Math 109 Spring 2022 Terminology

#### Updated 5/17/22

The following terms are taken from your textbook, "An Introduction to Mathematical Reasoning: numbers, sets and functions" by Peter J. Eccles. The terms are organized by chapter.

For each term, you should be able to 1) supply a definition, 2) exhibit an example, and, when appropriate, 3) exhibit a non-example. The vocabulary is fundamental and forms the foundation of this subject.

• Part I: Mathematical statements and proofs
• Chapter 1: The language of mathematics
• proposition
• predicate
• free variable(s)
• logical connective
• or, disjunction
• and, conjunction
• negation
• truth table
• Chapter 2: Implications
• implication
• hypothesis, antecedent
• conclusion, consequent
• universal implication
• existence statement
• P ⇒ Q
• If P, then Q.
• P implies Q.
• Q if P.
• P only if Q.
• Q whenever P.
• P is sufficient for Q.
• Q is necessary for P.
• converse
• P ⇔ Q
• P is equivalent to Q.
• P is necessary and sufficient for Q.
• P if and only if Q.
• P precisely when Q.
• field axioms
• commutativity
• associativity
• distributivity
• zero, additive identity
• unity, multiplicative identity
• subtraction, additive inverse
• division, multiplicative inverse
• Chapter 3: Proofs
• direct proof
• order axioms
• trichotomy law
• addition law
• multiplication law
• transitive law
• proof by cases
• constructing proofs backwards
• Chapter 4: Proof by contradiction
• non-existence result
• contradiction
• proving negative statements by contradiction
• proving implications by contradiction
• contrapositive
• proof by contrapositive
• Notes:
• A proof by contradiction is also called an indirect proof.
• A proof by contrapositive is considered to be a direct proof.
• Chapter 5
• the induction principle
• base case
• inductive step
• inductive hypothesis
• definition by induction
• n!, n factorial, factorial n
• the strong induction principle
• Fibonacci numbers
• Binet formula
• Part II
• Chapter 6
• set
• elements, members, points
• ∈, is an element of
• conditional definition of a set
• constructive definition of a set
• equality of sets
• ∅, empty set
• ⊆, is a subset of
• ⊂, is a proper subset of
• ⋂, intersection
• disjoint
• ⋃, union
• A - B, difference of A and B, complement of B in A
• P(X), power set of X
• Ac, complement of A
• associativity
• commutativity
• distributivity
• DeMorgan laws
• complementation
• double complement
• Chapter 7
• universal statement; universal quantifier
• existential statement; existential quantifier
• negation of a universal statement
• disproving statements of the form ∀ a ∈ A, P(a)
• negation of an existential statement
• disproving statements of the form ∃ a ∈ A, P(a)
• Cartesian product, X × Y
• ordered pair, coordinates
• Chapter 8
• function
• image, value
• domain
• codomain
• constant function
• identity function
• formula
• modulus function (absolute value function)
• equal, f = g
• restriction of f to A
• composite
• commutative (or not)
• sequence
• null sequence
• image (of a function)
• graph (of a function)
• Chapter 9
• injection, injective
• surjection, surjective
• bijection, bijective
• pre-image
• invertible
• inverse (function)
• Functions and subsets
• Peano's axioms
• successor function
• Part III
• Chapter 10
•   cardinality of X,   ∣ X ∣
•   finite set
•   infinite set
•   the addition principle
•   the multiplication principle
•   the inclusion-exclusion principle
• Chapter 11
•   the pigeonhole principle
•   minimum element of a set A ⊆ ℝ
•   maximum element of a set A ⊆ ℝ
• Chapter 12
•   Fun(X,y), the set of functions from X to Y
•   Inj(X,y), the set of injections from X to Y
•   permutation
•   cardinality of P(X), the power set of X
•   r-subset
•   Pr(X), the set of r-subsets of X
•   binomial coefficient
•   the binomial theorem
• Chapter 13
•
• Chapter 14
•   equipotent
•   denumerable, cardinality &alef sym;0
•   countable
•   uncountable
•   Dedekind's theorem
•   ℚ is denumerable
•   ℝ is uncountable
•   the continuum hypothesis
•   ∣ X ∣ < ∣ P(X) ∣
• Part IV
• Chapter 15
• The division theorem
• quotient
• remainder
• existence theorem
• uniqueness theorem
• Chapter 16
• greatest common divisor, gcd(a,b)
• Euclidean algorithm
• Lamé's theorem*
• Chapter 17
• integral linear combination
• greatest common divisor (alternative definition)
• coprime
• Part V
• Chapter 19
• congruent modulo m
• reflexive property
• symmetric property
• transitive property
• remainders modulo m, residues modulo m
• remainder map
• Chapter 20
• linear congruence
• diophantine equation
• Chapter 21
• congruence classes modulo m
• congruence class of a modulo m,   [a]m
• set of congruence classes modulo m,   m
• arithmetic of congruence classes
• addition of elements of m
• subtraction of elements of m
• multiplication of elements of m
• arithmetic of remainders
• addition of elements of Rm
• subtraction of elements of Rm
• multiplication of elements of Rm
• [a]m invertible,   a invertible modulo m
• inverse of [a]m,   inverse of a modulo m
• Chapter 22
•   partition
•   relation, a ∼ b
•   reflexive
•   symmetric
•   equivalence relation
•   equivalence class of a, [a]

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