1. Concept of Sets
A set is a well-defined collection of distinct objects. Sets are usually denoted by capital letters (A, B, C...) and their elements by small letters (a, b, x...).
📝 Representation of Sets
Roster/Tabular Form: All elements are listed, separated by commas, and enclosed within braces { }. E.g., Vowels = {a, e, i, o, u}
Set-builder Form: All elements possess a single common property. E.g., A = {x : x is a natural number < 6}
2. Types of Sets
- Empty/Null/Void Set (∅): Contains no elements.
- Singleton Set: Contains exactly one element.
- Finite vs Infinite: A set that consists of a definite number of elements is finite, otherwise infinite.
- Equal Sets: Two sets A and B are equal if they have exactly the same elements.
- Subsets: A is a subset of B (A ⊆ B) if every element of A is also an element of B. The total number of subsets of a set containing n elements is 2ⁿ.
- Power Set P(A): The collection of all subsets of set A.
- Universal Set (U): The basic set containing all elements and of which all other sets are subsets.
3. Operations on Sets
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Principle of Inclusion-Exclusion
Union (A ∪ B): Elements belonging to A, or B, or both.
Intersection (A ∩ B): Elements common to both A and B. Disjoint sets have no common elements (A ∩ B = ∅).
Difference (A - B): Elements belonging to A but not to B.
Complement (A'): Elements in the Universal set U that are not in A. A' = U - A.
De Morgan's Laws
1. (A ∪ B)' = A' ∩ B'
2. (A ∩ B)' = A' ∪ B'