7.1 Functions Defined on General Sets

Definitions

Function f from a set X to a set Y, denoted f: X → Y, is a relation from X, the domain of f, to Y, the co-domain of f, that satisfies two properties: (1) every element X is related to some element in Y, and (2) no element in X is related to more than one element in Y.
Range of f = image of X under f = {y ∈ Y | y = f(x), for some x in X}
Inverse image of y = {x ∈ X | f(x) = y}

If F: X → Y and G: X → Y are functions, then F = G if and only if F(x) = G(x) for every x ∈ X

Identity Function: I_X(x) = x for each x in X
Multiply Function: M(a, b) = a*b for each a in R and for each b in R
Hamming Distance Function: Let S_n be the set of all strings of 0's and 1's of length n. Define H: S_n × S_n → Z^nonneg as follows: For each pair of strings (s, t) ∈ S_n × S_n, H(s,t) = the number of positions where s and t have different values.
(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}. In other words, f: {0,1}ⁿ → {0,1}.

Let X = {1, 3, 5} and Y = {s, t, u, v}. Define f: X → Y by the following set of ordered pairs {(1, v), (3, s), (5, v)}.

Define a function on the set of nonnegative integers that can be used to define this sequence: 1, -1/3, 1/5, -1/7, 1/9, ...

Define F: Z × Z → Z × Z as follows: For every ordered pair (a,b) of integers, F(a,b) = (2a+1, 3b-2). Find F(2,1).

Find the Hamming Distance H(10101, 00011)

Draw an arrow diagram for the following boolean function: F(0,0) = 1, F(0,1) = 0, F(1,0) = 1, F(1,1) = 0

Show that g: Q → Z defined by the rule g(m/n) = m - n for all integers m and n with n ≠ 0 is not a function.