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40 Cards in this Set
- Front
- Back
_____ is the mathematics of counting and arranging objects, and _____ is the study of laws concerning the measurement of random or chance events. |
Combinatorics, probability |
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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 _______. |
domain, range |
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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.) |
some other thing also has to be true |
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A set may be specified using the set-rosternotation by writing all of its elements between ____. |
braces For example: {1, 2, 3} denotes the set whose elements are 1, 2, and 3. |
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a statement is "universal" because it says that theobject: |
satisfies a certain property for ALL things of a certain kind. |
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A universal statement says that a certain property is ____ for _____ _____ in a set.(For example: All positive numbers are greater than zero.) |
true for all elements in a set |
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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. |
structures |
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every integer is a real number, and because the integers are all separated from each other, the set of integers is called ____. |
discrete |
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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. |
existential statement |
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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. |
ordered |
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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 |
the Cartesian product of A and B |
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How many elements are in the set {1, {1}}? |
The set {1, {1}} has two elements: 1 and the set whose only element is 1. |
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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. |
subset |
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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 |
element |
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If S is a set, the notation ______ means that x is an element of S. |
x ∈ S |
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Let A = {1, 2, 3} and B = {u, v}.Find A × B |
A × B = {(1, u), (2, u), (3, u), (1, v), (2, v), (3, v)} |
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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. |
proper |
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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. |
variables |
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Probably the most important goal of a first course in discrete mathematics is to help students develop the ability to think ____. |
abstractly |
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Q often represents what in mathematics? |
Q = set of all rational numbers, or quotients of integers |
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R often represents what in mathematics? |
R = set of all real numbers |
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a statement is "existential" because it asserts that a certain object: |
exists |
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Symbol used to represent RELATION:For example, showing a relationship between elements x of set A, and elements y of set B: |
R example: x R y |
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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. |
the axiom of extension |
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The notation ____ means that x is not an element of S. |
x∈/ S (the cross goes through the ∈ symbol) |
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The phrases A is _____ in B and B _____ A are alternative ways of saying that A is a subset of B. |
contained, contains |
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The real number line is called _____ because it is imagined to have no holes. |
continuous |
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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 ____. |
real, origin |
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The squaring function is defined as ________________. The successor function is defined as: ________________. A constant function always outputs a _____. |
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. |
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The symbol ". . ." is called an ____ and is read |
ellipsis, "and so forth." |
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There is a positive integer that is less than or equal to every positive integer.This is an example of a ____ statement. |
Existential Universal Statement |
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Three of the most important kinds of sentences in mathematics are ____ statements, ____ statements, and ____ statements. |
universal statements,conditional statements, and existential statements. |
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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. |
recursively |
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Two _____ pairs (a, b) and (c, d) are equal if, and only if, a = c and b = d. |
ordered |
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We think of "____" as simply a collection of elements.@ |
sets |
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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 ______. |
co-domain |
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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. |
computer |
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Z often represents what in mathematics? |
Z = set of all integers |
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{x | P(x)}This is an example of ____ ____ notation. The | symbol is pronounced: "_____ _____" |
Set-builder notation,"such that" |
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The | symbol is read as: |
"such that" |