Lecture 10/06
Detailed Study Notes on Summation and Cardinality
Topic Overview
Today's topic is the summation of patient values.
Definition of Summation
By definition, the summation notation is represented as follows:
\Sigma{i=1}^n ai = a1 + a2 + a3 + … + an
Where:
i: index
n: upper limit
The sum ranges over the specified index from the lower limit to the upper limit.
Comparison with Calculus
The summation notation is similar to integral notation seen in calculus.
Integral notation, \int, can be thought of as an elongated version of summation.
Summation deals with discrete values while integrals handle continuous values.
Changing Limits of Summation
It is sometimes possible to change the lower and upper limits in a summation.
Example: For summing from 0 to 4, \Sigma{i=0}^4 ai implies:
a0 + a1 + a2 + a3 + a_4
Random set example:
Let the set be (S = {1, 3, 5}).
The summation \Sigma{i \in S} ai represents:
a1 + a3 + a_5
Introduction to Product Notation
Besides summation, there is also product notation, given as:
\Pi{i=1}^m ai = a1 \cdot a2 \cdot … \cdot a_m
Example of Summation
Consider the example:
\Sigma_{n=1}^{100} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{100}
Arithmetic and Geometric Progression
Recap on arithmetic and geometric sequences:
For an arithmetic progression \Sigma{i=1}^n ai where:
sn = \frac{(a1 + a_n) \cdot n}{2}
a1: first term
an: nth term
n: number of terms
For a geometric progression:
The formula is:
sn = a1 \frac{1 - r^n}{1 - r} (if r \neq 1)
Where:
r: common ratio
a1: first term
Properties of Summation
The summation notation is linear, which means:
You can break down the summation and factor out constants:
\Sigma{i=1}^m (ai + b) = \Sigma{i=1}^m ai + \Sigma_{i=1}^m b
A constant can be factored out:
\Sigma{i=1}^m (c \cdot ai) = c \cdot \Sigma{i=1}^m ai
Double Summation
Introducing double summation notation:
Example of how to calculate:
\Sigma{i=1}^3 \Sigma{j=2}^3 \frac{1}{i \cdot j}
You can fix one variable and sum over the other.
Working through each 'i' value:
When i=1: Sum over j=2 to 3
When i=2: Continue with fixed i=2 and sum over j
Keep track of the outside / inside indices to avoid mixing them up.
Examples of Double Summation
Example calculations:
Suppose, \Sigma{i=1}^3 \Sigma{j=1}^2 \frac{1}{i + j}
For i=1 and j increments:
\frac{1}{1+1} + \frac{1}{1+2}
Continue for i=2 and i=3
Summation Shapes
There are different geometrical shapes formed by summations:
Triangular Shape: Summation where the upper limit depends on the outer loop's index.
Rectangular Shape: Both indices have fixed limits and maintain a constant.
Cardinality of Sets
Cardinality defines the number of elements in a set, using notation such as ( |A| ) for set A.
Infinite sets require complex definitions beyond just saying they are infinite.
Example: Proper subsets can still have matching cardinalities if they are infinite.
Key Assertions About Sets:
Two sets have the same cardinality if a bijection exists between them.
A set is countably infinite if you can match its elements with natural numbers.
Bijection and Infinite Sets
Example of Infinite Sets: The set of even numbers is countably infinite because we can establish a bijection:
For natural numbers, all positive integers can be matched with even integers using a defined bijection.
Rational Numbers
The set Q represents rational numbers.
Even though they are densely populated on the number line, rational numbers have the same cardinality as integers.
Constructing a list of rational numbers ultimately establishes their countability.
Proving Uncountability of Real Numbers
Introduced by Cantor’s diagonal argument that proofs real numbers (especially intervals) are not countable.
Can be demonstrated through contradiction where a new number is constructed that cannot be included in any assumed list, thus showing the initial assumption was incorrect.