Diploma engineering Maths notes

Set Theory



Introduction to Set Theory
Set Theory is the branch of mathematics that deals with the study of sets — collections of
well-defined and distinct objects. It provides a fundamental basis for various branches of
mathematics including algebra, calculus, topology, and computer science.

Example: The collection of vowels in English can be written as V = {a, e, i, o, u}.

Definition and Representation of Sets
A set is a collection of distinct elements, represented either by listing elements within
braces or by describing properties shared by its members.

Sets can be represented in three ways:

1. **Roster Form:** Listing all elements explicitly. Example: A = {1, 2, 3, 4, 5}

2. **Set-builder Form:** Describing a property that elements satisfy.

Example: A = {x | x is a natural number less than 6}

3. **Venn Diagram:** Graphical representation of sets using circles.




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Types of Sets (with Examples)
1. Empty Set (∅): Contains no elements. Example: A = {x | x is an even prime number greater
than 2} = ∅
2. Finite Set: Has countable elements. Example: B = {2, 4, 6, 8, 10}
3. Infinite Set: Has uncountably many elements. Example: C = {x | x ∈ N}
4. Equal Sets: Two sets are equal if they have identical elements. Example: {1, 2, 3} = {3, 2, 1}
5. Universal Set: Contains all objects under consideration. Example: U = {1, 2, 3, 4, 5, 6}