Functions Defined on General Sets

Lecture 35

Section 7.1

Fri, Mar 30, 2007

Relations

A relation R from a set A to a set B is a subset of A B.

If x A and y B, then x has the relation R to y if (x, y) R.

Examples: Relations

Let A = B = R and let x, y R. Define R on A B to mean that y = x2.

Describe the elements of R. Define R on A B to mean that y < x2.

Describe the elements of R.

Functions

Let A and B be sets. A function from A to B is a relation from A

to B with the property that for every x A, there exists exactly one y B such that (x, y) f.

Write f : A B and f(x) = y. A is the domain of f. B is the co-domain (or range) of f.

Functions

Note that functions and algebraic expressions are two different things.

For example, do not confuse the algebraic expression (x + 1)2 with the function f : R R defined by f(x) = (x + 1)2.

Examples: Functions

f : R R by f(x) = x2. g : R R R by g(x, y) = 1 – x – y. h : R R R R by h(x, y) = (-x, -y). For any set A, k : (A) (A) (A) by

k(X, Y) = X Y. For any sets A and B, m : (A) (B) by

m(X) = X B.

Inverse Images

If f(x) = y, we say that y is the image of x and that x is an inverse image of y.

The inverse image of y is the set

f -1(y) = {x X | f(x) = y}.

Inverse Images

In the previous examples, findf -1(4).g-1(0).m-1({a}), where A = {a, b, c}, B = {a, b}.

Equality of Functions

Let f : X Y and g : X Y be two functions.

Then f = g if f(x) = g(x) for all x X.

Equality of Functions

Are f(x) = |x| and g(x) = x2 equal? Are f(x) = 1 and g(x) = sec2 x – tan2 x

equal? Are f(x) = log x2 and g(x) = 2 log x equal?

Another Example

Earlier we saw that a subset of a universal set could be represented as a binary string.

For example,U = {a, b, c, d} 1111A = {a, b} 1100 = {} 0000

Describe this as a function.

Well Defined

A function is well defined if for every x in the domain of the function, there is exactly one y in the codomain that is related to it.

Well Defined

Why are the following “functions” not well defined?f : Q Z, f(a/b) = a.g : Z Z Q, g(a, b) = a/b.h : Q Z Z, h(a/b) = (a, b).k : Q Q, k(a/b) = b/a.

Can they be “repaired?”

Boolean Functions

A Boolean function is a function whose domain is {0, 1} … {0, 1} (or {0, 1}n) and codomain is {0, 1}.

Example: Let p, q be Boolean variables and define f(p, q) = p q.

p q f(p, q)

1 1 1

1 0 0

0 1 0

0 0 0

The Number of Boolean Functions

How many Boolean functions are there in 2 variables?What are they?

How many Boolean functions are there in 3 variables?

How many Boolean functions are there in n variables?

Boolean Functions

What Boolean function is defined byf(x, y) = xy?

What Boolean function is defined byf(x, y) = x + y – xy?

What Boolean function is defined byf(x) = 1 – x?

What Boolean function is defined byf(x, y, z) = 1 – xy – z + xyz?