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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.
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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\}.
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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 $C$.