1. Functions

2. Definition and notation
Definition: A function f from a set X to a set Y
is a relationship between elements of X and Y
with the property that
each element of X is related to a unique element of Y.
Denoted f:X→Y .
X is called domain of f; Y is called co-domain of f.
Example: X=Z, Y=Z and f : x → 2∙x / 2
xX ! yY such that f(x)=y .
f(x) is called f of x (or image of x under f).
Range of f = {yY | y=f(x) for some x in X}
Inverse image of y = {xX | f(x)=y}
Example(cont.): range of f = all even integers ;
inverse image of 4 = {3, 4} .

3. Examples of Functions Squaring function: f : x → x2 .
Constant function: f : x → 3 .
Linear function: f : x → 3x+2 .
Factorial function: f : n → n! .
Any sequence can be considered
as a function defined on a set of integers.
E.g., sequence 2,5,8,11,14,…
is a function from Z+ to Z+
defined as follows f : n → 3n-1

4. Boolean Functions Recall the truth tables:
Can be considered as a function;
the domain is the set
of all ordered couples of 0 and 1;
the co-domain is {0,1} .
Input
Output p q
p  q 1

5. Boolean Functions Definition:
An (n-place) Boolean function 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}.
Example: f : (x,y,z) → (~x  y)  z

6. One-to-one Functions Definition:
Let F be a function from set X to set Y.
F is one-to-one (or injective) iff
for all elements x1, x2  X
if F(x1)=F(x2) then x1=x2 .
Examples: Define f : Z → Z by f(n)=2n+3 ;
g : R → R by f(x)=x2 .
Then f is one-to-one, and g is not.

7. Onto Functions Definition: Let F be a function from set X to set Y.
F is onto (or surjective) iff
for any element y  Y there is a x X
such that F(x)=y .
Examples: Define f : Z → Z by f(n)=2n+3 ;
g : Z → Z by f(n)=n-2 .
Then g is onto, and f is not.

8. Exponential Functions
The exponential function with base b
is the following function from R to R+ :
expb(x) = bx
b0=1 b-x = 1/bx
bubv = bu+v
(bu)v = buv
(bc)u = bucu

9. Logarithmic Functions
The logarithmic function with base b
(b>0, b1)
is the following function from R+ to R:
logb(x) = the exponent to which b must raised to obtain x .
Symbolically, logbx = y  by = x .
Properties:

10. One-to-one Correspondences
Definition: A one-to-one correspondence
(or bijection) from a set X to a set Y
is a function f:X→Y
that is both one-to-one and onto.
Examples:
1) Linear functions: f(x)=ax+b when a0
(with domain and co-domain R)
2) Exponential functions: f(x)=bx (b>0, b1)
(with domain R and co-domain R+)
3) Logarithmic functions: f(x)=logbx (b>0, b1)
(with domain R+ and co-domain R)

11. Inverse Functions Theorem:
Suppose F: X→Y is a one-to-one correspondence.
Then there is a function F-1: Y→X defined as follows:
Given any element in Y,
F-1(y) = the unique element x in X
such that F(x)=y .
The function F-1 is called the inverse function for F.
Example:
The logarithmic function with base b (b>0, b 1)
is the inverse of the exponential function with base b.