1. Functions Defined on General Sets
Lecture 30
Section 7.1
Fri, Mar 4, 2005

2. 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.
We may also write x R y.

3. Examples: Relations
Let A = B = R. Let x, y  R. Define x R y to mean that y = x2.
Describe R.
Let A = B = R. Let x, y  R. Define x R y to mean that y < x2.
Is R  R a relation?
Is  a relation?

4. 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 of f.

5. 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.

6. 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.

7. Examples: Functions
Let n be the size of a complete binary tree. Define f : N  R by f(n) = the average number of nodes visited to locate a randomly selected value in the tree.
We found earlier that if n = 2k – 1, then
f(n)  k – 1.
In general, what expression approximately describes the values of this function?

8. 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}.
In the previous examples, find
f -1(4).
g-1(0).
h-1(5, 10).

9. 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.
Are the functions f(x) = |x| and g(x) = x2 equal?
Are the functions f(x) = 1 and g(x) = sec2 x – tan2 x equal?

10. Boolean Functions
A Boolean function is a function whose domain is {0, 1}  …  {0, 1} 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

11. 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?

12. Boolean Functions What Boolean function is defined by f(x, y) = xy?
f(x, y) = x + y – xy?
f(x) = 1 – x?
f(x, y, z) = 1 – xy – z + xyz?