A set is a well-defined collection of distinct objects. "Well-defined" means there is no ambiguity about whether an object belongs to the collection. The objects in a set are called its elements or members.
Notation: Sets are usually denoted by capital letters (A, B, C …) and elements by small letters. If x is an element of set A, we write x ∈ A; if not, x ∉ A.
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Key Concepts
- Representation of Sets
- Roster/Tabular form: List all elements inside curly braces. E.g., A = {1, 2, 3, 4}.
- Set-builder form: Describe a property. E.g., B = {x : x is a natural number, x < 5}.
- Types of Sets
- Empty set (Null set) Ø: A set with no elements. E.g., {x : x2 = -1, x ∈ R}.
- Finite set: Has a countable number of elements. E.g., {2, 4, 6, 8}.
- Infinite set: Has unlimited elements. E.g., N = {1, 2, 3, …}.
- Singleton set: Exactly one element. E.g., {0}.
- Universal set (U): The set containing all objects relevant to a discussion.
Subsets: A is a subset of B (A ⊆ B) if every element of A is also in B. Every set is a subset of itself; Ø is a subset of every set. A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.
Power Set: The set of all subsets of A, written P(A). If |A| = n, then |P(A)| = 2n.
- Operations on Sets
- Union: A ∪ B = {x : x ∈ A or x ∈ B}
- Intersection: A ∩ B = {x : x ∈ A and x ∈ B}
- Difference: A − B = {x : x ∈ A and x ∉ B}
- Complement: A' = {x : x ∈ U and x ∉ A}
Venn diagrams visually represent these operations.
Cardinality formula: |A ∪ B| = |A| + |B| − |A ∩ B|
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Worked Examples
Write A = {x : x is a vowel in the English alphabet} in roster form.
A = {a, e, i, o, u}
If A = {1, 2, 3} and B = {2, 3, 4, 5}, find A ∪ B and A ∩ B.
A ∪ B = {1, 2, 3, 4, 5}; A ∩ B = {2, 3}
List all subsets of {p, q}.
Subsets: Ø, {p}, {q}, {p, q}. So |P({p,q})| = 22 = 4.
In a class of 40 students, 25 play cricket, 20 play football, and 10 play both. How many play at least one sport?
|C ∪ F| = 25 + 20 − 10 = 35.
If U = {1,2,3,4,5,6,7,8}, A = {2,4,6,8}, find A'.
A' = {1,3,5,7}
Show that A − B = A ∩ B'.
x ∈ A − B means x ∈ A and x ∉ B, i.e., x ∈ A and x ∈ B'. So A − B = A ∩ B'.
If |A| = 15, |B| = 12, |A ∩ B| = 5, find |A ∪ B|.
|A ∪ B| = 15 + 12 − 5 = 22.
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Common mistakes
- Confusing Ø (empty set) with {0} (set containing zero — this has one element).
- Forgetting that Ø ⊆ every set and every set is a subset of itself.
- Miscounting subsets: always use 2n.
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Summary
A set is a well-defined collection. Key operations — union, intersection, difference, complement — follow specific laws (commutative, associative, distributive, De Morgan's). The cardinality formula is essential for solving practical counting problems.