Prove that two functions f and g are equal if and only if...

Prove that two functions f and g are equal if and only if

1. The domain of f is equal to the domain of g

and

2. for all x that are elements of the domain of f, f(x)=g(x)

The domain simply refers to the first coordinate for all coordinates in the function.

Proof:
Assume f=g

1. Suppose that x is an element in the domain of f, then (x,y) is an element of f for some y. Since we assume that f=g then (x,y) is also an element of g and x must be an element in the domain of g. Thus the domain of f must be a subset of the domain of g, and similarly g must be a subset of f. So the two domains are equal.

2. Again suppose that x is an element of the domain of f. Then for some y, (x,y) is an element of f. Since f=g, (x,y) must also be an element of g. Therefore f(x)=y=g(x)

Characteristic Functions

Characteristic functions define the functions of sets and the functions of all object outside the set.

Take for consideration the set A that is part of the Universe. Then the characteristic function can be seen as

f(x)={1 if x is an element of A, and 0 if x is an element of everything but A, (or U-A)}.

Ranges and Codomains

Ranges and Codomains

The domain of a function can be seen as the first coordinate of an ordered pair, and the range cam be seen as the second coordinate. Thus for the ordered pair (3,5) 3 is the domain, and 5 is the domain. Every function can only have one domain and one range. Even though functions can have many many ordered pairs.

There can be, however, many co domains, as long as the function is a subset of the codomain. Consider the example:

A = {4,5,6}
B = {2,5,7}

Then the following two sets are functions of A and B with different codomains.

f1={(4,2),(5,5),(6,7)}
f2={(4,5),(5,7),(6,5)}

Functions

Function as a word was first used by Leibniz in 1694, and Euler popularized the notation f(x) in 1734. In terms of intuitive logic a function describes a rule of correspondence between two sets (i.e.: distance as a function of time).

Looking at functions in terms of relations a function f from A to B is a relation from A to B such that if (x,y) is an element in f and (x,z) is an element in f, then y = z. And also, the Dom(f) = A.