Cs 225 - Ch. 1

Front 1

_____ is the mathematics of counting and arranging objects, and _____ is the study of laws concerning the measurement of random or chance events.

Back 1

Combinatorics, probability

Front 2

A _____ is known as a set of "inputs", and a ______ is known as a set of "outputs". The elements of the ______ are "mapped" to the elements of the _______.

Back 2

domain, range

Front 3

A conditional statement says that:If one thing is true then _________________________.

(For example: If 378 is divisible by 18, then 378 is divisible by 6.)

Back 3

some other thing also has to be true

Front 4

A set may be specified using the set-rosternotation by writing all of its elements between ____.

Back 4

braces

For example: {1, 2, 3} denotes the set whose elements are 1, 2, and 3.

Front 5

a statement is "universal" because it says that theobject:

Back 5

satisfies a certain property for ALL things of a certain kind.

Front 6

A universal statement says that a certain property is ____ for _____ _____ in a set.(For example: All positive numbers are greater than zero.)

Back 6

true for all elements in a set

Front 7

Discrete mathematical _____ are the abstract _____ thatdescribe, categorize, and reveal the underlying relationships among discrete mathematical objects. Those studied in this book are the sets of integers and rational numbers, general sets, Boolean algebras, functions, relations, graphs and trees, formal languages and regular expressions, and finite-state automata.

Back 7

structures

Front 8

every integer is a real number, and because the integers are all separated from each other, the set of integers is called ____.

Back 8

discrete

Front 9

Given a property that may or may not be true, an _____ statement says thatthere is at least one thing for which the property is true. (For example: There is a prime number that is even.

Back 9

existential statement

Front 10

Given elements a and b, the symbol (a, b) denotes the ____ pair consisting of aand b together with the specification that a is the first element of the pair and b is thesecond element.

Back 10

ordered

Front 11

Given sets A and B, A × B = {(a, b)| a∈A and b∈B}Also read "A cross B", is called the ______ product of A and B

Back 11

the Cartesian product of A and B

Front 12

How many elements are in the set {1, {1}}?

Back 12

The set {1, {1}} has two elements: 1 and the set whose only element is 1.

Front 13

If A and B are sets, then A is called a ____ of B, written A⊆B, if, and only if,every element of A is also an element of B.

Back 13

subset

Front 14

If C is the set of all countries that are currently in the United Nations, then the United States is an ____ of C, and if I is the set of all integers from 1 to 100, then the number 57 is an ____ of I

Back 14

element

Front 15

If S is a set, the notation ______ means that x is an element of S.

Back 15

x ∈ S

Front 16

Let A = {1, 2, 3} and B = {u, v}.Find A × B

Back 16

A × B = {(1, u), (2, u), (3, u), (1, v), (2, v), (3, v)}

Front 17

Let A and B be sets. A is a _____ subset of B if, and only if, every element of A is in B but there is at least one element of B that is not in A.

Back 17

proper

Front 18

One of the most important reasons for using _____ in mathematics is that it givesyou the ability to refer to quantities unambiguously throughout a lengthy mathematicalargument, while not restricting you to consider only specific values for them.

Back 18

variables

Front 19

Probably the most important goal of a first course in discrete mathematics is to help students develop the ability to think ____.

Back 19

abstractly

Front 20

Q often represents what in mathematics?

Back 20

Q = set of all rational numbers, or quotients of integers

Front 21

R often represents what in mathematics?

Back 21

R = set of all real numbers

Front 22

a statement is "existential" because it asserts that a certain object:

Back 22

exists

Front 23

Symbol used to represent RELATION:For example, showing a relationship between elements x of set A, and elements y of set B:

Back 23

R

example: x R y

Front 24

The axiom of _____ says that a set is completely determined by what its elementsare—not the order in which they might be listed or the fact that some elements might belisted more than once.

Back 24

the axiom of extension

Front 25

The notation ____ means that x is not an element of S.

Back 25

x∈/ S

(the cross goes through the ∈ symbol)

Front 26

The phrases A is _____ in B and B _____ A are alternative ways of saying that A is a subset of B.

Back 26

contained, contains

Front 27

The real number line is called _____ because it is imagined to have no holes.

Back 27

continuous

Front 28

The set of ____ numbers is usually pictured as the set of all points on a line, and the number 0 corresponds to a middle point, called the ____.

Back 28

real, origin

Front 29

The squaring function is defined as ________________.

The successor function is defined as: ________________.

A constant function always outputs a _____.

Back 29

Squaring function: replaces x with x^2 .

Successor function: replaces x with x + 1.

A constant function always outputs a constant number, no matter the input.

Front 30

The symbol ". . ." is called an ____ and is read

Back 30

ellipsis, "and so forth."

Front 31

There is a positive integer that is less than or equal to every positive integer.This is an example of a ____ statement.

Back 31

Existential Universal Statement

Front 32

Three of the most important kinds of sentences in mathematics are ____ statements, ____ statements, and ____ statements.

Back 32

universal statements,conditional statements, and existential statements.

Front 33

To think ____ means to address a problem by assuming that similar problems of a smaller naturehave already been solved and figuring out how to put those solutions together to solvethe larger problem.

Back 33

recursively

Front 34

Two _____ pairs (a, b) and (c, d) are equal if, and only if, a = c and b = d.

Back 34

ordered

Front 35

We think of "____" as simply a collection of elements.@

Back 35

sets

Front 36

When relating set's A and B (A R B), the set A is called the domain of R and the set B is called its ______.

Back 36

co-domain

Front 37

Whereas the ideas of calculus were fundamental to the science and technology of theindustrial revolution, the ideas of discrete mathematics underlie the science and technology of the ____ age.

Back 37

computer

Front 38

Z often represents what in mathematics?

Back 38

Z = set of all integers

Front 39

{x | P(x)}This is an example of ____ ____ notation. The | symbol is pronounced: "_____ _____"

Back 39

Set-builder notation,"such that"

Front 40

The | symbol is read as:

Back 40

"such that"