sets
SET THEORY – FULL STUDY NOTES
1. Definition of a Set
A set is a well-defined collection of distinct objects.
Well-defined → clear membership rules
Distinct → no repetition
Examples:
A = {1, 2, 3}
B = {red, blue, green}
2. Set Notation
a \in A → element of A
a \notin A → not an element of A
3. Types of Sets
(a) Empty Set
No elements
or { }
(b) Finite Set
Limited elements
Example: {2, 4, 6}
(c) Infinite Set
Unlimited elements
Example: {1, 2, 3, …}
(d) Equal Sets
Same elements
Order does not matter
{1,2,3} = {3,2,1}
4. Universal Set
The set containing all elements under discussion.
Symbol: U
5. Subsets, Proper Subsets & Supersets
(a) Subset
A is a subset of B if all elements of A are in B.
A \subseteq B
Example:
A = {1,2}, B = {1,2,3} → A ⊆ B
✔ A set is always a subset of itself.
(b) Proper Subset
A is a proper subset of B if:
A ⊆ B
A ≠ B
A \subset B
Example:
{1,2} ⊂ {1,2,3}
(c) Superset
B is a superset of A if B contains all elements of A.
B \supseteq A
Example:
{1,2,3} ⊇ {1,2}
(d) Proper Superset
B is a proper superset of A if:
B contains A
B ≠ A
B \supset A
Example:
{1,2,3} ⊃ {1,2}
6. Set Operations
(a) Union
A ∪ B = all elements in A or B
Example:
A = {1,2}, B = {2,3}
A ∪ B = {1,2,3}
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(b) Intersection
A ∩ B = common elements
A ∩ B = {2}
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(c) Difference
A − B = elements in A not in B
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(d) Complement
A′ = elements in U not in A
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7. Cardinality
Number of elements in a set
• n(A) = size of A
Example:
A = {1,2,3} → n(A) = 3
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8. Key Formula
For two sets:
n(A \cup B) = n(A) + n(B) - n(A \cap B)
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9. Power Sets
Definition:
The power set of A is the set of all possible subsets of A.
• Notation: P(A)
Example:
A = {1,2}
P(A) = { {}, {1}, {2}, {1,2} }
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Important Rule:
If a set has n elements:
\text{Number of subsets} = 2^n
Example:
A = {1,2,3} → 2^3 = 8 subsets
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10. Key Insight
• Subset = relationship between two sets
• Power set = collection of all subsets of one set
• Proper subset = strict inclusion
• Superset = reverse of subset