Discrete Mathematics For AIML - Chapter 1 Sets, Relations & Functions

Chapter 1 — Sets, Relations & Functions

This chapter covers foundational concepts of sets, relations, and functions. These concepts are essential for understanding data structures, feature mappings, and relationships in AI/ML datasets.

1.1 Sets

A set is a collection of distinct elements. Sets are the building blocks of mathematical data representation. Key Concepts:

Subset: A set where all elements belong to another set.
Power set: The set of all subsets of a given set.
Universal set: Contains all elements under consideration.

AI/ML Context: Representing features, labels, and datasets as sets helps in organizing and processing data efficiently.

1.2 Set Operations

Fundamental operations to combine or compare sets.

Union ( ∪ ): Combines all elements from two sets.
Intersection ( ∩ ): Elements common to both sets.
Difference ( − ): Elements in one set but not the other.
Complement: Elements not in the given set relative to the universal set.

AI/ML Context: Useful in feature selection, filtering datasets, and logical conditions for ML preprocessing.

1.3 Relations

A relation defines how elements from one set relate to elements of another set.

Reflexive: Every element relates to itself.
Symmetric: If a relates to b, then b relates to a.
Transitive: If a relates to b and b relates to c, then a relates to c.
Equivalence relations: Relations that are reflexive, symmetric, and transitive.

AI/ML Context: Equivalence relations can group similar data points, helping in clustering and data segmentation.

1.4 Functions

A function maps each element from an input set to exactly one element in an output set.

One-to-one (Injective): Every input maps to a unique output.
Onto (Surjective): Every element in the output set has a pre-image in the input set.
Bijective: Both one-to-one and onto; establishes a perfect pairing between input and output sets.

AI/ML Context: Functions represent transformations, mappings from input features to predictions, and data preprocessing pipelines.

1.5 Practical Python Examples

# Representing sets in Python
features = {'height', 'weight', 'age'}
labels = {'healthy', 'sick'}

# Set operations
all_elements = features.union(labels)
common = features.intersection(labels)
difference = features.difference(labels)

# Functions as mappings
def feature_to_label(height, weight):
    if height > 170 and weight < 70:
        return 'healthy'
    else:
        return 'sick'

print(all_elements, common, difference)
print(feature_to_label(180, 65))

1.6 Why Sets, Relations & Functions Matter in AI/ML

Understanding sets, relations, and functions helps structure datasets, define relationships between variables, and implement feature transformations. These concepts form the backbone for:

Feature engineering and preprocessing
Mapping inputs to outputs in predictive models
Clustering and equivalence-based grouping

1.7 Exercises

Define a universal set of 10 features and create subsets for training and testing.
Implement a function that maps numerical features to a categorical label.
Create two sets of labels and compute their union, intersection, and difference.

Hints / Solutions

Use Python set() and operations like union(), intersection(), difference().
Define a Python function using def and if-else statements.
Use set1.union(set2), set1.intersection(set2), set1.difference(set2).

1.8 Further Reading & Videos

Discrete Mathematics and Its Applications – Kenneth H. Rosen
Khan Academy – Set Theory and Functions
YouTube – "Discrete Mathematics for Computer Science" by WilliamFiset

Next Chapter: Logic & Propositional Calculus — Learn to reason, represent, and evaluate logical statements, crucial for AI/ML decision-making.

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