Chapter 1: Set Theory

Lecture Notes on Set Theory

Topics: Definition of Set, Types of Sets, Set Operations, Theorems, Addition Theorem, Detailed Solved Examples, Applications in Biology/Biotechnology/Neuroscience

Table of Contents

1. Definition of Set
2. Types of Sets
3. Union and Intersection
4. Addition Theorem
5. De Morgan's Laws
6. Solved Examples
7. Applications in Biology / Biotechnology / Neuroscience
8. Practice Exercises
9. Summary

1. Definition of Set

2. Types of Sets

3. Union and Intersection

4. Addition Theorem

5. De Morgan's Laws

6. Solved Examples

7. Applications in Biology / Biotechnology / Neuroscience

8. Practice Exercises

9. Summary

1. Definition of Set

A set is a well-defined collection of distinct objects.

Example: $A = \{1,2,3,4\}$ is a set of natural numbers.

Applications:

• Genes expressed in a cell can be represented as a set.
• Neurons activated under a stimulus form a set.
• Proteins in a pathway form a set.
• Species in an ecosystem form a set.

2. Types of Sets

• Empty Set: $\varnothing$
• Singleton Set: $\{a\}$
• Finite Set: $\{1,2,3\}$
• Infinite Set: $\{1,2,3,\dots\}$

3. Union and Intersection

Union: $$A \cup B = \{x : x \in A \text{ or } x \in B\}$$ Intersection: $$A \cap B = \{x : x \in A \text{ and } x \in B\}$$

Union Venn Diagram
(A ∪ B includes both circles)

Intersection Venn Diagram
(A ∩ B includes only overlap)

4. Addition Theorem

For two sets: $$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

Proof of Addition Theorem

When we count $n(A)+n(B)$, the common elements in $A\cap B$ are counted twice. Therefore subtract them once: $$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

5. De Morgan's Laws

$$(A\cup B)' = A' \cap B'$$ $$(A\cap B)' = A' \cup B'$$

Proof of De Morgan's First Law

Let $x \in (A\cup B)'$. Then $x \notin A\cup B$. So $x \notin A$ and $x \notin B$. Hence $x \in A' \cap B'$. Therefore, $$(A\cup B)' = A' \cap B'$$

6. Solved Examples

Example 1

If $A=\{1,2,3\}$ and $B=\{3,4,5\}$, find $A\cup B$ and $A\cap B$. $$A\cup B=\{1,2,3,4,5\}$$ $$A\cap B=\{3\}$$

Example 2

If $n(A)=20$, $n(B)=15$, and $n(A\cap B)=5$, find $n(A\cup B)$. Using: $$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$ $$n(A\cup B)=20+15-5=30$$

Example 3

If $U=\{1,2,3,4,5\}$ and $A=\{1,3,5\}$, find $A'$. $$A'=\{2,4\}$$

Example 4

If $A=\{2,4,6\}$ and $B=\{4,6,8\}$, then: $$A-B=\{2\}$$

Example 5

In a survey: - 25 students study Biology - 20 study Biotechnology - 8 study both Find total students studying at least one. $$n(A\cup B)=25+20-8=37$$

7. Applications in Biology / Biotechnology / Neuroscience

• Biology: Union of gene sets from two species.
• Biotechnology: Intersection of proteins in two pathways.
• Neuroscience: Overlap of neurons activated by two stimuli.
• Bioinformatics: Complement of disease biomarker sets.

8. Practice Exercises

Exercise 1

If $A=\{1,2\}$ and $B=\{2,3\}$, find $A\cup B$ and $A\cap B$.

Exercise 2

Verify: $$(A\cup B)' = A' \cap B'$$

Exercise 3

If $n(A)=30$, $n(B)=18$, $n(A\cap B)=6$, find $n(A\cup B)$.

9. Summary

Set theory is used to classify, compare, and analyze overlapping datasets in mathematics and biological sciences.