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A universal statement says that a certain property is true for all elements in a set. (For example: All positive numbers are greater than zero.)
A conditional statement says that if one thing is true then some other thing also has to be true. (For example: If 378 is divisible by 18, then 378 is divisible by 6.)
Given a property that may or may not be true, an existential statement says that there is at least one thing for which the property is true. (For example: There is a prime number that is even.)
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Set Roster Notation
If S is a set, the notation x \in S means that x is an element of S. The
notation x \notin S means that x is not an element of S. A set may be
specified using the set-roster notation by writing all of its elements
between braces. For example, \{1, 2, 3\} denotes the set whose elements are
1, 2, and 3. A variation of the notation is sometimes used to describe a
very large set, as when we write \{1, 2, 3, \dots, 100\} to refer to the set
of all integers from 1 to 100. A similar notation can also describe an
infinite set, as when we write \{1, 2, 3, \dots\} to refer to the set of all
positive integers. (The symbol \dots is called an ellipsis and is read
"and so forth.")
The axiom of extension says that a set is completely determined by what its elements are - not the order in which they might be listed or the fact that some elements might be listed more than once.
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| Symbol | Set |
|---|---|
| \mathbb{R} | the set of all real numbers |
| \mathbb{Z} | the set of all integers |
| \mathbb{Q} | the set of all rational numbers, or quotients of integers |
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Set-Builder Notation
Let S denote a set and let P(x) be a property that elements of S may or
may not satisfy. We may define a new set to be the set of all elements x in
S such that P(x) is true. We denote this set as follows:
\{x \in S | P(x)\} Where x is "the set of all" and | is "such that."
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Definition
If A and B are sets, then A is called a subset of B, written
A \subseteq B, if and only if, every element of A is also an element of B.
Symbolically:
A \subseteq B means that for every element x, if x \in A then x \in B.
The phrases A is contained in B and B contains A are alternative
ways of saying that A is a subset of B.
It follows from the definition of subset that for a set A not to be a subset
of a set B means that there is at least one element of A that is not an
element of B. Symbolically:
A \nsubseteq B means that there is at least one element x such that
x \in A and x \notin B.
Definition
Let A and B be sets. A is a proper 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.
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Notation
Given elements a and b, the symbol (a, b) denotes the ordered pair
consisting of a and b together with the specification that a is the first
element of the pair and b is the second element. Two ordered pairs (a, b)
and (c, d) are equal if, and only if, a = c and b = d. Symbolically:
(a, b) = (c, d) means that a = c and b = d.
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Definition
Let n be a positive integer and let x_1, x_2, \dots, x_n be (not necessarily
distinct) elements. The ordered $n$-tuple, $(x_1, x_2, \dots, x_n)$,
consists of x_1, x_2, \dots, x_n together with the ordering: first
x_1, then x_2, and so forth up to x_n. An ordered 2-tuple is called an
ordered pair, and an ordered 3-tuple is called an ordered triple.
Two ordered $n$-tuples (x_1, x_2, \dots, x_n) and (y_1, y_2, \dots, y_n) are
equal if, and only if, x_1 = y^1, x^2 = y^2, \dots, and x_n = y_n.
Symbolically:
(x_1, x_2, \dots x_n) = (y_1, y_2, \dots, y_n) \leftrightarrow x_1 = y_1, x^2 = y_2, \dots , x_n = y_n Page 35
Definition
Given sets A_1, A_2, \dots, A_n, the Cartesian product of
A_1, A_2, \dots, A_n denoted A_1 \times A_2 \times \dots \times A_n, is the
set of all ordered $n$-tuples (a_1, a_2, \dots, a_n) where
a_1 \in A_1, a_2 \in A_2, \dots , a_n \in A_n.
Symbolically:
A_1 \times A_2 \times \dots \times A_n = \{(a_1, a_2, \dots, a_n) | a_1 \in A_1, a_2 \in A_2, \dots , a_n \in A_n\} In particular,
A_1 \times A_2 = \{(a_1, a_2) | a_1 \in A_1 \text{ and } a_2 \in A_2\} is the Cartesian product of A_1 and A_2.
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Definition
Let n be a positive integer. Given a finite set A, a string of length n
over $A$ is an ordered $n$-tuple of elements of A written without
parentheses or commas. The elements of A are called the characters of the
string. The null string over A is defined to be the "string" with no
characters. It is often denoted \lambda and is said to have length 0. If
A = \{0, 1\}, then a string over A is called a bit string.
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Definition
Let A and B be sets. A relation R from A to $B$ is a subset of
A \times B. Given an ordered pair (x, y) in A \times B, **x is related
to y by R, written xRy, if, and only if, (x, y) is in R. The set A
is the domain of R and the set B is called its co-domain.
The notation for a relation R may be written symbolically as follows:
xRy means that (x, y) \in R.
The notation x\cancel{R}y means that x is not related to y by R:
x\cancel{R}y means that (x, y) \notin R.
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Definition
A function F form a set A to a set $B$ is a relation with domain A and
co-domain B that satisfies the following two properties:
-
For every element
xinA, there is an elementyinBsuch that(x, y) \in F. -
For all elements
xinAandyandzinB,
\text{if } \quad (x, y) \in F \text{ and } (x, z) \in F \text{, } \quad \text{ then } \quad y = z Page 42
Function Notation
If A and B are sets and F is a function from A to B, then given any
element x in A, the unique element in B that is related to x by F is
denoted F(x), which is read "F of x."
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Definition
A graph G consists of two finite sets: a nonempty set V(G) of
vertices and a set E(G) of edges, where each edge is associated with a
set consisting of either one or two vertices called its endpoints. The
correspondence from edges to endpoints is called the edge-endpoint function.
An edge with just one endpoint is called a loop, and two or more distinct edges with the same set of endpoints are said to be parallel. An edge is said to connect its endpoints; two vertices that are connected by an edge are called adjacent; and a vertex that is an endpoint of a loop is said to be adjacent to itself.
An edge is said to be incident on each of its endpoints, and two edges incident on the same endpoint are called adjacent. A vertex on which no edges are incident is called isolated.
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Definition
A directed graph, or digraph, consists of two finite sets: a nonempty
set V(G) of vertices and a set D(G) of directed edges, where each is
associated with an ordered pair of vertices called its endpoints. If edge
e is associated with the pair (v, w) of vertices, then e is said to be the
(directed) edge from v to w.
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Definition
Let G be a graph and v a vertex of G. The degree of $v$, denoted
$deg(v)$, equals the number of edges that are incident on v, with an edge
that is a loop counted twice.