A Set is a well-defined collection of distinct objects. The objects in a set are called its members or elements. "Well-defined" means that for any object, we can clearly determine whether it belongs to the collection or not.
There are two primary ways to represent or describe a set:
Sets are categorized based on the number and nature of their elements:
| Set Type | Description | Example |
|---|---|---|
| Empty / Null Set | A set containing no elements, denoted by { } or Ø. | {x : x is a person living on Mars} |
| Finite Set | A set with a countable number of elements. | {1, 2, 3, 4, 5} |
| Infinite Set | A set with uncountable elements. | The set of all points on a line. |
| Equal Sets | Two sets having exactly the same elements. | A={1,2}, B={2,1} |
| Equivalent Sets | Sets with the same number of elements (Cardinality). | A={a,b}, B={1,2} |
If every element of set A is also an element of set B, then A is called a Subset of B (denoted as A ⊆ B).
Power Set: The collection of all possible subsets of a set A is called the Power Set of A, denoted as P(A).
If a set has n elements, the total number of subsets (elements in the power set) is 2ⁿ.
Example: If A = {1, 2}, then P(A) = {Ø, {1}, {2}, {1, 2}}. Total elements = 2² = 4.
Venn diagrams are graphical representations of sets using closed curves (usually circles) within a rectangle representing the Universal Set (U).
They are essential for visualizing the relationships between different sets and verifying set identities visually.
Common operations used to combine or modify sets include:
These laws (identities) are used to simplify set expressions:
Q: Is {0} an empty set?
A: No. It is a singleton set containing one element: zero. An empty set has zero elements and is written as { }.
Q: What is a Disjoint Set?
A: Two sets A and B are disjoint if their intersection is empty (A ∩ B = Ø).