Chapter 2 - Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Question 1

set

Answer 1

a collection of distinct objects

Question 2

axiom

Answer 2

a basic assumption of a theory

Question 3

paradox

Answer 3

a logical inconsistency

Question 4

element, a member of a set

Answer 4

an object in a set

Question 5

roster method

Answer 5

a method that describes a set by listing its elements

Question 6

set builder notation

Answer 6

the notation that describes a set by stating a property an element must have to be a member

Question 7

∅ (empty set, null set)

Answer 7

the set with no members

Question 8

universal set

Answer 8

the set containing all objects under consideration

Question 9

Venn diagram

Answer 9

a graphical representation of a set or sets

Question 10

S = T (set equality)

Answer 10

S and T have the same elements

Question 11

S ⊆ T (S is a subset of T)

Answer 11

every element of D is also an element of T

Question 12

S ⊂ T (S is a proper subset of T)

Answer 12

S is a subset of T and S ≠ T

Question 13

finite set

Answer 13

a set with n elements, where n is a nonnegative integer

Question 14

infinite set

Answer 14

a set that is not finite

Question 15

|S| (the cardinality of S)

Answer 15

the number of elements in S

Question 16

P(S) (the power set of S)

Answer 16

the set of all subsets of S

Question 17

A ∪ B (the union of A and B)

Answer 17

the set containing those elements that are in at least one of A and B

Question 18

A ∩ B (the intersection of A and B)

Answer 18

the set containing those elements that are in both A and B

Question 19

A — B (the difference of A and B)

Answer 19

the set containing those elements that are in A but not in B

Question 20

Ᾱ (the complement of A)

Answer 20

the set of elements in the universal set that are not in A

Question 21

A ⊕ B (the symmetric difference of A and B)

Answer 21

the set containing those elements in exactly one of A and B

Question 22

membership table

Answer 22

a table displaying the membership of elements in sets

Question 23

function from A to B

Answer 23

an assignment of exactly one element of B to each element of A

Question 24

domain of ƒ

Answer 24

the set A, where ƒ is a function from A to B

Question 25

codomain of ƒ

Answer 25

the set B, where ƒ is a function from A to B

Question 26

b is the inage of a under ƒ

Answer 26

b = ƒ(a)

Question 27

a is a pre-image pf b under ƒ

Answer 27

ƒ(a) = b

Question 28

range of ƒ

Answer 28

the set of images of ƒ

Question 29

onto function, surjection

Answer 29

a function from A to B such that every element of B is the image of some element in A

Question 30

one-to-one function, injection

Answer 30

a function such that the images of elements in its domain are distinct

Question 31

one-to-one correspondence, bijection

Answer 31

a function that is both one-to-one and onto

Question 32

inverse of ƒ

Answer 32

the function that reverses the correspondence given by ƒ (when ƒ is a bijection)

Question 33

ƒ ○ g (composition of ƒ and g)

Answer 33

the function that assigns ƒ(g(x)) to x

Question 34

⎿x⏌ (floor function)

Answer 34

the largest integer not exceeding x

Question 35

⎾x⏋(ceiling function)

Answer 35

the smallest integer greater than or equal to x

Question 36

partial function

Answer 36

an assignment to each element in a subset of the domain a unique element in the codomain

Question 37

sequence

Answer 37

a function with domain that is a subset of the set of integers

Question 38

geometric progression

Answer 38

a sequence in the form a, ar, ar², . . . , where a and r are real numbers

Question 39

arithmetic progression

Answer 39

a sequence in the form a, a + d, a + 2d, . . . , where a and d are real numbers

Question 40

string

Answer 40

a finite sequence

Question 41

empty string

Answer 41

a string of length zero

Question 42

recurrence relation

Answer 42

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

Question 43

Σⁿₓ₌₁ aₓ

Answer 43

the sum of a₁ + a₂ + . . . + aₙ

Question 44

Πⁿₓ₌₁ aₓ

Answer 44

the product of a₁a₂ . . . aₙ

Question 45

cardinality

Answer 45

two sets A and B have the same cardinality if there is a one-to-one correspondence from A to B

Question 46

countable set

Answer 46

a set that either is finite or can be placed in one-to-one correspondence with the set of positive integers

Question 47

uncountable set

Answer 47

a set that is not countable

Question 48

ℵ₀ (aleph null)

Answer 48

the cardinality of a countable set

Question 49

ς

Answer 49

the cardinality of the set of real numbers

Question 50

Cantor diagonalization argument

Answer 50

a proof technique used to show that the set of real numbers is uncountable

Question 51

computable function

Answer 51

a function for which there is a computer program in some programming language that finds its values

Question 52

uncomputable function

Answer 52

a function for which no computer program in a programming language exists that finds its values

Question 53

continuum hypothesis

Answer 53

the statement that no set A exists such that ℵ₀ < |A| < ς