Set Theory

Introduction

• Set theory is a branch of mathematical logic that deals with the study of sets, which are collections of objects.

• Sets are fundamental in mathematics and provide a foundation for various mathematical concepts.

Basic Concepts

1. Set it is a collection of items

   A set is a well-defined collection of distinct objects, called elements, represented by listing the elements inside curly braces.

   • Example: A = {1, 2, 3} represents a set A with elements 1, 2, and 3.

2. Subset: A set A is said to be a subset of set B if every element of A is also an element of B.

   • Example: A = {1, 2, 3} is a subset of B = {1, 2, 3}.

     Improper Subset

3. Union

4. The union of two sets A and B, denoted by A ∪ B, is the set that contains all the elements that are in A or B (or both).

   • Example: A = {1, 2} and B = {2, 3} ⇒ A ∪ B = {1, 2, 3}.

5. proper subset: A is a proper subset of B if all the member of A area also members of B but B has additional elements which are not in A

   Example: A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6, 7}

6. improper subset An improper subset is a subset that includes all the elements of the original set, as well as possibly additional elements.

   It is denoted by the symbol ⊆ (subset or equal to).

7. For example, if we have a set A = {1, 2, 3} and a set B = {1, 2, 3, 4}, then A is an improper subset of B because A includes all the elements of B.

8. Intersection: The intersection of two sets A and B, denoted by A ∩ B, is the set that contains all the elements that are in both A and B.

   • Example: A = {1, 2} and B = {2, 3} ⇒ A ∩ B = {2}.

9. Complement: The complement of a set A, denoted by A', is the set that contains all the elements that are not in A.

   1 2 3

   • Example: A = {1, 2} ⇒ A' = {3}.

10. Universal Set: The universal set, denoted by U , is the set that contains all the elements under consideration.

11. Empty Set: The empty set, denoted by ∅ or {}, is the set that contains no elements.

12. Set difference set of all elements in A which are not in B

13. A(1,2,3) B(2,3,4,5)

    set diff(1)

Questions and Answers

1. What is a set?

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

2. What is a subset?

   • A set A is a subset of set B if every element of A is also an element of B.

3. What is the union of two sets?

   • The union of two sets A and B is the set that contains all the elements that are in A or B

     Venn Diagram

     • A Venn diagram is a graphical representation of sets using circles or other shapes. It is named after the English mathematician John Venn.