The Logic of Groups: Set Theory
Set theory is a branch of mathematical logic that studies sets, which are collections of objects.
Sets and Notation
A set is a well-defined collection of distinct objects, considered as an object in its own right. The objects are called elements.
Notation: Sets are usually denoted by capital letters, and elements are listed within curly braces {}.
Example: A = {1, 2, 3, 4}
Element of: The symbol ∈ means 'is an element of'. So, 3 ∈ A.
Empty Set (∅): A set with no elements.
Set Operations
1.Union (∪): The union of two sets A and B is the set of all elements that are in A, or in B, or in both.
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
2.Intersection (∩): The intersection of two sets A and B is the set of all elements that are in both A and B.
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.
3.Complement (A'): The complement of a set A is the set of all elements in the universal set (U) that are not in A.
Example: If U = {1, 2, 3, 4, 5} and A = {1, 2}, then A' = {3, 4, 5}.
Venn Diagrams
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets.
Each set is typically represented by a circle.
The universal set is represented by a rectangle enclosing the circles.
The overlapping region of two circles represents the intersection of the two sets.
The total area covered by two circles represents the union of the two sets.
Venn diagrams are a powerful visual tool for understanding and solving problems involving sets.