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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.