6.8 KiB
Page 449
Definition
A function f from a set X to a set $Y$, denoted: f: X \to Y, is a
relation from X, the domain of f, to Y, the co-domain of f, that
satisfies two properties: (1) every element in X is related to some element in
Y, and (2) no element in X is related to more than one element in Y. Thus,
given any element x in X, there is a unique element in Y that is related
to x by f. If we call this element y, then we say that "f sends x to
$y$" or "f maps x to $y$" and write x \xrightarrow{f} y or f: x \to y.
The unique element to which f sends x is denoted
f(x) and is called f of x, or the output of f for the input x, or the
value of f at x, or the image of x under f.
The set of all values of f taken together is called the range of $f$ or the
image of X under $f$. Symbolically:
\text{range of } f = \text{ image of } X \text{ under } f = \{y \in Y | y = f(x), \text{ for some } x \text{ in } X\} Given an element y in Y, there may exist elements in X with y as their
image. When x is an element such that f(x) = y, then x is called a
preimage of $y$ or an inverse image of $y$. The set of all inverse images
of y is called the inverse image of $y$. Symbolically:
\text{ the inverse image of } y = \{x \in X | f(x) = y\} Page 451
Theorem 7.1.1 A Test for Function Equality
If F: X \to Y and G: X \to Y are functions, then F = G if, and only if,
F(x) = G(x) for every x \in X.
Proof:
Suppose F: X \to Y and G: X \to Y are functions; that is, F and G are
relations from X to Y that satisfy the two additional function properties.
Then F and G are subsets of X \times Y, and for (x, y) to be in F
means that y is the unique element related to x by F, which we denote as
F(x). Similarly, for (x, y) to be in G means that y is the unique
element related to x by G, which we denote as G(x).
Now suppose that F(x) = G(x) for every x \in X. Then if x is any element
of X,
(x, y) \in F \Leftrightarrow y = F(x) \Leftrightarrow y = G(x) \Leftrightarrow (x, y) \in G because F(x) = G(x).
So F and G consist of exactly the same elements and hence F = G.
Conversely, if F = G, then for every x \in X,
y = F(x) \Leftrightarrow (x, y) \in F \Leftrightarrow (x, y) \in G \Leftrightarrow y = G(x) because F and G consist of exactly the same elements.
Thus, since both F(x) and G(x) equal y, we have that
F(x) = G(x) Page 453
Definition Logarithms and Logarithmic Functions
Let b be a positive real number with b \neq 1. For each positive real number
x, the logarithm with base b of $x$, written \log_bx, is the exponent
to which b must be raised to obtain x. Symbolically:
\log_bx = y \Leftrightarrow b^y = x The logarithmic function with base $b$ is the function from \mathbb{R}^+
to \mathbb{R} that takes each positive real number x to \log_bx.
Page 455
Definition
An ($n$-place) Boolean function f is a function whose domain is the set of
all ordered $n$-tuples of $0$'s and $1$'s and whose co-domain is the set
\{0, 1\}. More formally, the domain of a Boolean function can be described as
the Cartesian product of n copies of the set \{0, 1\}, which is denoted
\{0, 1\^n}. Thus f: \{0, 1\}^n \to \{0, 1\}.
Page 457
Definition
If f: X \to Y is a function and A \subseteq X and C \subseteq Y, then
f(A) = \{y \in Y | y = f(x) \text{ for some } x \text{ in } A\} and
f^{-1}(C) = \{x \in X | f(x) \in C\} f(A) is called the image of $A$, and f^{-1}(C) is called the inverse
image of $C$.
Page 463
Definition
Let F be a function from a set X to a set Y. F is one-to-one (or
injective) if, and only if, for all elements x_1 and x_2 in X,
\text{if } F(x_1) = F(x_2) \text{, then } x_1 = x_2 or, equivalently,
\text{if } x_1 \neq x_2 \text{, then } F(x_1) \neq F(x_2) Symbolically:
F: X \to Y \text{ is one-to-one } \Leftrightarrow \forall x_1, x_2 \in X \text{, if } F(x_1) = F(x_2) \text{ then } x_1 = x_2 Page 466
Definition: Hash Function
A hash function is a function defined from a larger, possibly infinite, set of data to a smaller fixed-size set of integers.
Page 469
Definition
Let F be a function from a set X to a set Y. F is onto (or
surjective) if, and only if, given any element y in Y, it is possible to
find an element x in X with the property that y = F(x).
Symbolically:
F:X \to Y \text{ is onto } \Leftrightarrow \forall y \in Y, \exists x \in X \text{ such that } F(x) = y Page 472
Laws of Exponents
If b and c are any positive real numbers and u and v are any real
numbers, the following laws of exponents hold true:
7.2.1
b^ub^v = b^{u + v} 7.2.2
(b^u)^v = b^{uv} 7.2.3
\frac{b^u}{b^v} = b^{u - v} 7.2.4
(bc)^u = b^uc^u Page 473
Theorem 7.2.1 Properties of Logarithms
For any positive real numbers b, c, x and y with b \neq 1 and
c \neq 1 and for every real number a:
a. \log_b(xy) = \log_bx + \log_by
b. \log_b\left(\dfrac{x}{y}\right) = \log_bx - \log_by
c. \log_b(x^a) = a\log_bx
d. \log_cx = \dfrac{\log_bx}{\log_bc}
Page 475
Definition
A one-to-one correspondence (or bijection) from a set X to a set Y
is a function F: X \to Y that is both one-to-one and onto.
Page 478
Theorem 7.2.2
Suppose F: X \to Y is a one-to-one correspondence; in other words, suppose F
is one-to-one and onto. Then there is a function F^{-1}: Y \to X that is
defined as follows:
Given any element y in Y,
F^{-1}(y) = \text{ that unique element } x \text{ in } X \text{ such that } F(x) \text{ equals } y Or, equivalently,
F^{-1}(y) = x \Leftrightarrow y = F(x) Page 478
Definition
The function F^{-1} of Theorem 7.2.2 is called the inverse function for
F.
Page 479
Theorem 7.2.3
If X and Y are sets and F: X \to Y is one-to-one and onto, then
F^{-1}:Y \to X is also one-to-one and onto.
Proof:
F^{-1} is one-to-one:
Suppose y_1 and y_2 are elements of Y such that
F^{-1}(y_1) = F^{-1}(y_2). [We must show that y_1 = y_2.] Let
x = F^{-1}(y_1) = F^{-1}(y_2). Then x \in X, and by definition of F^{-1},
F(x) = y_1 \text{ since } x = F^{-1}(y_1) and
F(x) = y^2 \text{ since } x = F^{-1}(y_2) Consequently, y_1 = y_2 because each is equal to F(x). [This is what was to
be shown.]
F^{-1} is onto:
Suppose x \in X. [We must show that there exists an element y in Y such
that F^{-1}(y) = x.] Let y = F(x). Then y \in Y, and by definition of
F^{-1}, F^{-1}(y) = x [as was to be shown.]