Unit 1.1: Sets

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1. Introduction to Sets

Set Theory was developed by Georg Cantor and is foundational in mathematics. Sets are used to define relations and functions in various areas of mathematics.

2. Definition of a Set

A Set is a well-defined collection of distinct objects.

Example: The set of vowels in the English alphabet can be written as A = {a, e, i, o, u}

3. Methods of Representation of Sets

Tabular (Roster) Method: Lists all elements inside curly braces {}.

Example: The set of even numbers less than 10 is B = {2, 4, 6, 8}

Set Builder Method: Describes a property that the elements satisfy.

Example: The set of all positive integers less than 5 can be written as C = {x: x is a positive integer less than 5}, which represents C = {1, 2, 3, 4}

4. Types of Sets

Empty Set (Null Set or Void Set): A set with no elements, denoted by ∅ or {}.

Example: The set of all natural numbers less than 0 is an empty set, D = ∅

Singleton Set: A set with exactly one element.

Example: The set containing only the number 7 is E = {7}

Finite Set: A set with a finite number of elements.

Example: The set of prime numbers less than 10 is F = {2, 3, 5, 7}

Infinite Set: A set with an infinite number of elements.

Example: The set of all natural numbers is G = {1, 2, 3, 4, …}

Equal Sets: Two sets are equal if they contain exactly the same elements.

Example: If H = {1, 2, 3} and I = {3, 2, 1}, then H = I

Equivalent Sets: Two sets are equivalent if they have the same number of elements.

Example: J = {a, b, c} and K = {1, 2, 3} are equivalent because both have 3 elements.

Universal Set: The set that contains all elements under consideration, usually denoted by U.

Example: If the universal set is the set of all natural numbers, then U = {1, 2, 3, …}

Power Set: The set of all subsets of a given set X, denoted by P(X).

Example: If L = {a, b}, then P(L) = {∅, {a}, {b}, {a, b}}

5. Subsets and Supersets

Subset: A set M is a subset of a set N (denoted M ⊆ N) if every element of M is also an element of N.

Example: If M = {1, 2} and N = {1, 2, 3}, then M ⊆ N

Superset: A set O is a superset of P (denoted O ⊇ P) if P ⊆ O.

Example: N ⊇ M from the previous example.

Proper Subset: A set Q is a proper subset of R (denoted Q ⊂ R) if Q ⊆ R and Q ≠ R.

Example: If Q = {1, 2} and R = {1, 2, 3}, then Q ⊂ R

6. Venn Diagrams

Venn Diagrams represent sets graphically, with sets depicted as circles within a universal set.

Example: If S = {1, 2} and T = {2, 3}, a Venn diagram can show the intersection S ∩ T = {2} and the union as S ∪ T = {1, 2, 3}

7. Operations on Sets

Union of Sets (A ∪ B): The set of all elements that are in A, or in B, or in both.

Example: If U = {1, 2, 3} and V = {3, 4, 5}, then U ∪ V = {1, 2, 3, 4, 5}

Intersection of Sets (A ∩ B): The set of all elements that are in both A and B.

Example: If U = {1, 2, 3} and V = {3, 4, 5}, then U ∩ V = {3}

Difference of Sets (A − B): The set of elements that are in A but not in B.

Example: If U = {1, 2, 3} and V = {3, 4, 5}, then U − V = {1, 2}

Complement of a Set (A′): The set of all elements in the universal set U that are not in A.

Example: If U = {1, 2, 3, 4, 5} and W = {1, 2}, then the complement of W is W′ = {3, 4, 5}

8. Important Points to Remember

A Subset is a set where all elements are contained in another set.

Example: {a, b} ⊆ {a, b, c}

The Power Set of a set with n elements has 2n subsets.

Example: For X = {a, b}, P(X) has 22 = 4 subsets.

The Universal Set contains all elements under consideration in a given context.

Example: If we consider all integers, the universal set might be U = {…, −2, −1, 0, 1, 2, …}

The Empty Set is a subset of every set.

Example: ∅ ⊆ {1, 2, 3}