Chapter 2 - Basic Structures: Sets, Functions, Sequences, Sums, and Matrices Flashcards
Excludes: 2.6 - Matrices
set
a collection of distinct objects
axiom
a basic assumption of a theory
paradox
a logical inconsistency
element, a member of a set
an object in a set
roster method
a method that describes a set by listing its elements
set builder notation
the notation that describes a set by stating a property an element must have to be a member
∅ (empty set, null set)
the set with no members
universal set
the set containing all objects under consideration
Venn diagram
a graphical representation of a set or sets
S = T (set equality)
S and T have the same elements
S ⊆ T (S is a subset of T)
every element of D is also an element of T
S ⊂ T (S is a proper subset of T)
S is a subset of T and S ≠ T
finite set
a set with n elements, where n is a nonnegative integer
infinite set
a set that is not finite
|S| (the cardinality of S)
the number of elements in S
P(S) (the power set of S)
the set of all subsets of S
A ∪ B (the union of A and B)
the set containing those elements that are in at least one of A and B
A ∩ B (the intersection of A and B)
the set containing those elements that are in both A and B
A — B (the difference of A and B)
the set containing those elements that are in A but not in B
Ᾱ (the complement of A)
the set of elements in the universal set that are not in A
A ⊕ B (the symmetric difference of A and B)
the set containing those elements in exactly one of A and B
membership table
a table displaying the membership of elements in sets
function from A to B
an assignment of exactly one element of B to each element of A
domain of ƒ
the set A, where ƒ is a function from A to B
codomain of ƒ
the set B, where ƒ is a function from A to B
b is the inage of a under ƒ
b = ƒ(a)
a is a pre-image pf b under ƒ
ƒ(a) = b
range of ƒ
the set of images of ƒ
onto function, surjection
a function from A to B such that every element of B is the image of some element in A
one-to-one function, injection
a function such that the images of elements in its domain are distinct
one-to-one correspondence, bijection
a function that is both one-to-one and onto
inverse of ƒ
the function that reverses the correspondence given by ƒ (when ƒ is a bijection)
ƒ ○ g (composition of ƒ and g)
the function that assigns ƒ(g(x)) to x
⎿x⏌ (floor function)
the largest integer not exceeding x
⎾x⏋(ceiling function)
the smallest integer greater than or equal to x
partial function
an assignment to each element in a subset of the domain a unique element in the codomain
sequence
a function with domain that is a subset of the set of integers
geometric progression
a sequence in the form a, ar, ar², . . . , where a and r are real numbers
arithmetic progression
a sequence in the form a, a + d, a + 2d, . . . , where a and d are real numbers
string
a finite sequence
empty string
a string of length zero
recurrence relation
a equation that expresses the nth term aₙ of a sequence in terms of one or more of the previous terms of the sequence for all ontegers n greater than a particular integer
Σⁿₓ₌₁ aₓ
the sum of a₁ + a₂ + . . . + aₙ
Πⁿₓ₌₁ aₓ
the product of a₁a₂ . . . aₙ
cardinality
two sets A and B have the same cardinality if there is a one-to-one correspondence from A to B
countable set
a set that either is finite or can be placed in one-to-one correspondence with the set of positive integers
uncountable set
a set that is not countable
ℵ₀ (aleph null)
the cardinality of a countable set
ς
the cardinality of the set of real numbers
Cantor diagonalization argument
a proof technique used to show that the set of real numbers is uncountable
computable function
a function for which there is a computer program in some programming language that finds its values
uncomputable function
a function for which no computer program in a programming language exists that finds its values
continuum hypothesis
the statement that no set A exists such that ℵ₀ < |A| < ς
