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:

  • $\Sigma<em>{i=1}^n a</em>i = a<em>1 + a</em>2 + a<em>3 + … + a</em>n$

  • 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, $\Sigma<em>{i=0}^4 a</em>i$ implies:

  • $a<em>0 + a</em>1 + a<em>2 + a</em>3 + a_4$

  • Random set example:

  • Let the set be (S = {1, 3, 5}).

  • The summation $\Sigma<em>{i \in S} a</em>i$ represents:

    • $a<em>1 + a</em>3 + a_5$

Introduction to Product Notation

• Besides summation, there is also product notation, given as:

  • $\Pi<em>{i=1}^m a</em>i = a<em>1 \cdot a</em>2 \cdot … \cdot a_m$

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 $\Sigma<em>{i=1}^n a</em>i$ where:

  • $s<em>n = \frac{(a</em>1 + a_n) \cdot n}{2}$

    • a1: first term

    • an: nth term

    • n: number of terms

  • For a geometric progression:

  • The formula is:

    • $s<em>n = a</em>1 \frac{1 - r^n}{1 - r}$ (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:

  • $\Sigma<em>{i=1}^m (a</em>i + b) = \Sigma<em>{i=1}^m a</em>i + \Sigma_{i=1}^m b$

  • A constant can be factored out:

  • $\Sigma<em>{i=1}^m (c \cdot a</em>i) = c \cdot \Sigma<em>{i=1}^m a</em>i$

Double Summation

• Introducing double summation notation:

  • Example of how to calculate:

  • $\Sigma<em>{i=1}^3 \Sigma</em>{j=2}^3 \frac{1}{i \cdot j}$

  • 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, $\Sigma<em>{i=1}^3 \Sigma</em>{j=1}^2 \frac{1}{i + j}$

  • 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.