Links:
Home
Calendar
Syllabus
Homework
PtTA
Terminology
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
- 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
- A^{c}, 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
- P_{r}(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, ℤ_{m}
- arithmetic of congruence classes
- addition of elements of ℤ_{m}
- subtraction of elements of ℤ_{m}
- multiplication of elements of ℤ_{m}
- arithmetic of remainders
- addition of elements of R_{m}
- subtraction of elements of R_{m}
- multiplication of elements of R_{m}
- [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]
Links:
Home
Calendar
Syllabus
Homework
PtTA
Terminology