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?
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