Sets in Mathematics

Sets

• Sets are collections of things (elements)
• Set notation uses curly brackets {} to enclose elements
• Elements are separated by commas
• Example: A = {1, 2, 3} or B = {Hello, Water, Jack}

Set Representation

• Elements' existence in a set can be represented

Set Equality

• Set equality is determined by the elements, not the order: {1, 2, 3} = {3, 2, 1}

Set-Builder Notation

• Describes a set by a rule of what qualifies as an element instead of listing all elements (useful for infinite sets).
• Example: {x | x ∈ R, x > 0} means all real numbers greater than zero

Number Sets

• Natural numbers (N): {1, 2, 3, ...}
• Integers (Z): {... -3, -2, -1, 0, 1, 2, 3 ...}
• Rational numbers (Q): numbers that can be expressed as a fraction of two integers
• Irrational numbers: numbers that cannot be expressed as a fraction of two integers (e.g., √2, π)
• Real numbers (R): the combination of rational and irrational numbers
• Imaginary numbers (i): the square root of -1
• Complex numbers: a combination of real and imaginary numbers

Types of Sets

• Universal set (U): a set containing all elements
• Empty set (∅ or {}): a set containing no elements; denoted by a pair of empty curly brackets.
• Singleton: a set containing a single element
• Subset: A set where all its elements are also in another set, denoted by "⊂".
• If all elements of A are in B, and B has more elements than A, then A is a proper subset of B, denoted by "⊂ ".

Set Cardinality

• Cardinality: the number of elements in a set, denoted by n(A)

Equivalent Sets

• Sets with the same number of elements are equivalent.

Subsets and Proper Subsets

• Subset (⊂) – a set containing all elements from another set, or equal to it, meaning all and possibly some more.
• Proper subset (⊂)- a set containing all elements from another set, meaning all and possibly some more, but with no duplicate elements

Ordered Pairs and n-Tuples

• Ordered pairs: pairs of elements where order matters ((a, b) ≠ (b, a)).
• n-tuples: generalization of ordered pairs to n elements.

Cartesian Product

• Cartesian product: a set of all possible ordered pairs from two sets.

Set Operations

• Union (A ∪ B): contains all elements in A or B.
• Intersection (A ∩ B): contains only elements that are in both A and B.
• Difference (A - B): contains elements in A but not in B.
• Complement (A'): contains elements in the universal set that are not in A.

Venn Diagrams

• Visual representation of set relationships and operations.