Functions: Surjective, Bijective, Injective


A function ffrom a set P to a set Q is an assignment of exactly one element of Q to each element of P.
We write f(p) = q
if q is the unique element of Q assigned by the function f to the element p of P.
If f is a function from P to Q, we write f: P→Q


Assume the following function f:  P→Q with
P = {Adam, Mariam, Omar, Lina}
Q = {London, Dubai, Istanbul, Paris}
f(Adam) = Paris
f(Mariam) = London
f(Omar) = Istanbul)
f(Lina) = Dubai
Here is the range of f is Q.
If we make the following changes:
f(Adam) = Dubai then f is still a function but the range is {Dubai, London, Istanbul}

Another way to represent f:


Injections, Surjections and Bijections

A function f: P→Q is said to be one-to-one (or injective), if and only if
∀x,  y∈P (f(x) = f(y) →  x=y)
Basically, f is one-to-one if and only if it does not map two distinct elements of P onto the same element of Q.


Is f one-to-one? f(Adam) = Dubai
f(Mariam) = London
f(Omar) = Istanbul)
f(Lina) = Dubai
The answer is no because Adam and Lina are mapped onto the same element of the image.

Properties of Functions


See the below function and check if it is injective, surjective, or bjiective.
Answer: f is neither.
Question: What if Omar is also connected to Dubai? Answer: Here, f is not even a function.
Question: What if Lina is also connected to Dubai? Answer: In that case, f is injective, surjective, as well as bijective.


Let f: O→P , g: Q→O. The composition of f with g, denoted f∘g, is the function from Q to P defined by
f∘g(x) = f(g(x))


let f(x) = x2 and g(x)=x ‒ 3 are functions from R to R
Find the compositions f∘g and g∘f
f∘g(x) = f(g(x)) = f(x ‒ 3) = (x ‒ 3)2
g∘f(x) = g(f(x)) = g(x2) = x2 ‒ 3
Which means that f∘g and g∘f are generally different.

Sequences and Summations

Definition:A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2, . . . } to a set S.
Note: the sets {0, 1, 2, 3, . . . , k} and {1, 2, 3, 4, . . . , k} are called initial segments of N.
Notation: if f is a function from {0, 1, 2, . . .} to S we usually denote f(i) by ai and we write.
where k is the upper limit (usually ∞).
Summation Notation


Example of Summation Notation

Geometric Progression

A geometric progression is a sequence of the form a, ar, ar2, ar3, ar4, . . . .
Try to prove that
Geometric Progression
(Try to figure out what it is if r = 1).
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Date of last modification: February 20, 2019