Contents

1. Set Theory and Logic

1. Set Theory

1.1 Need for Sets

1.2 Representation of Sets

1.3 Set Operations

2. Types of Sets

3. Logic

3.1 Propositional Logic

3.2 Propositional Equivalences

3.3 Rules of Inference

3.4 Application of Propositional Logic; Translating English Sentences

4. Induction

4.1 Principle of Induction

4.2 Applications

4.3 Strong Induction

2. Relations and Functions

1. Relations and their Properties

1.1 Relations on a Set

1.2 Properties of Relations

2. Partition of a Set

3. n-Ary Relations and their Applications

4. Representing Relations

4.1 Representing Relations using Matrices

4.2 Digraphs of Relations

4.3 Composition of Relations

5. Poset (Partially Ordered Set)

5.1 Partially Ordered Set (or Poset)

5.2 Comparibility

5.3 Totally Ordered Set

5.4 Hasse Diagram

6. Lattices

7. Closures of Relations

7.1 Reflexive Closure of R

7.2 Symmetric Closure of R

7.3 Transitive Closure and Warshall’s Algorithm

8. Functions

8.1 Types of Function

8.2 Invertible Functions

8.3 Composition of Functions

3. Counting and Probability

1. Basics of Counting

1.1 The Product Rule [or Multiplication Principle]

1.2 The Sum Rule [or Addition Principle]

2. Permutations and Combinations

2.1 Permutations

2.2 Combination

2.3 Binomial Coefficients and Identities

3. The Principle of Inclusion Exclusion

4. The Pigeonhole Principle

5. Generalized Permutations and Combinations

5.1 Permutation with Repetition

5.2 Combinations with Repetition

5.3 Permutations with Indistinguishable Objects

5.4 Distributing Objects into Boxes

6. Probability

6.1 Classical Definition of Probability

6.2 Axiomatic Approach to Probability

6.3 Theorems of Probability

6.4 Conditional Probability

6.5 Bayes’ Theorem

4. Data Presentation and Aggregation

1. Data Condensation

2. Types of Data (Primary and Secondary)

2.1 Primary and Secondary Data

2.2  Attributes and Variables

3. Data Presentation

3.1 Frequency Distribution

3.2 Histogram

3.3 Less than and More than Type Ogive curves

3.4 Bar Plots

4. Measure of Central Tendency

4.1 Arithmetic Mean

4.2 Weighted Arithmetic Mean

4.3 Median

4.4 Mode

4.5 Empirical Relationship among Mean, Median and Mode

4.6 Geometric Mean (GM)

4.7 Harmonic Mean (HM)

5. Partition Values    

5.1 Quartiles

5.2 Deciles

5.3 Percentiles

5.4 Box Plot

6. Measure of Dispersion

6.1 Variance

6.2 Standard Deviation (S.D.)

6.3 Properties of Variance and Standard Deviation

6.4 Coefficient of Variation

6.5 Root Mean Square (RMS)

5. Correlation Theory and Sampling

1. Introduction

2. Correlation

3. Scatter Diagram Method

4. Karl Pearson’s Coefficient of Correlation (r)

4.1 Properties of Correlation Coefficient

4.2 Interpretation of ‘r’

5. Correlation of Attributes

6. Regression

6.1 Linear Regression

6.2 Fitting of Straight Line Using Least Square Method

7. Elementary Sampling Theory

7.1 Population

7.2 Sample

7.3 Random Samples and Random Numbers