2.1 Set Operations and Properties

Set operations are the building blocks of set theory, allowing us to combine and manipulate sets in various ways. They're crucial for understanding relationships between different groups of elements and solving complex problems in mathematics and computer science.

This section covers basic operations like union and intersection, as well as more advanced concepts like power sets and De Morgan's laws. These tools help us analyze and work with sets efficiently, forming the foundation for more complex set theory applications.

Set Operations

Basic Set Operations

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  Set language and notation :: Maths View original

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• Union (A ∪ B) combines elements from two sets A and B, creating a new set containing all unique elements from both
• Intersection (A ∩ B) identifies common elements between two sets A and B, forming a new set with only shared elements
• Complement (A') includes all elements in the universal set that are not in set A
• Difference (A - B) creates a new set with elements from A that are not in B
• Symmetric difference (A △ B) produces a set containing elements in either A or B, but not in both

Visualizing Set Operations

• Venn diagrams graphically represent relationships between sets using overlapping circles
• Two-circle Venn diagrams illustrate basic operations like union and intersection
• Three-circle Venn diagrams depict more complex relationships among multiple sets
• Shading in Venn diagrams indicates specific regions corresponding to set operations (union shaded area covers both circles)

Properties of Set Operations

• Commutative property applies to union and intersection (A ∪ B = B ∪ A, A ∩ B = B ∩ A)
• Associative property holds for union and intersection ((A ∪ B) ∪ C = A ∪ (B ∪ C))
• Distributive property relates union and intersection (A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C))
• Identity elements exist for union (empty set) and intersection (universal set)
• Complement of complement returns the original set (A'' = A)

Set Relationships

Subset and Proper Subset

• Subset (A ⊆ B) indicates every element of A is also an element of B
• Proper subset (A ⊂ B) means A is a subset of B, but A ≠ B
• Empty set is a subset of all sets
• A set is always a subset of itself
• Subset notation uses symbols ⊆ (subset) and ⊂ (proper subset)

Power Set and Universal Set

• Power set P(A) contains all possible subsets of set A, including the empty set and A itself
• Number of elements in a power set equals 2^n, where n is the number of elements in the original set
• Universal set U encompasses all possible elements under consideration for a given context
• Complement of a set is defined relative to the universal set
• Universal set varies depending on the problem or situation being analyzed

Set Equality and Cardinality

• Two sets are equal if they contain exactly the same elements (A = B if A ⊆ B and B ⊆ A)
• Cardinality |A| represents the number of elements in set A
• Finite sets have a countable number of elements
• Infinite sets contain an unlimited number of elements (natural numbers, real numbers)
• Cardinality of the power set |P(A)| = 2^|A|

Set Representations and Laws

Set Notation and Representation

• Set-builder notation describes sets using a property or rule (A = {x | x is an even number less than 10})
• Roster notation lists all elements of a set within curly braces (B = {1, 2, 3, 4, 5})
• Empty set represented by {} or ∅
• Interval notation describes continuous ranges of numbers ([0, 1] represents all real numbers between 0 and 1, inclusive)

De Morgan's Laws

• First law states (A ∪ B)' = A' ∩ B'
• Second law states (A ∩ B)' = A' ∪ B'
• These laws relate complements of unions and intersections
• Useful for simplifying complex set expressions
• Can be extended to more than two sets ((A ∪ B ∪ C)' = A' ∩ B' ∩ C')

Distributive and Other Set Laws

• Distributive law of union over intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• Distributive law of intersection over union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
• Absorption laws: A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A
• Idempotent laws: A ∪ A = A and A ∩ A = A
• Complement laws: A ∪ A' = U and A ∩ A' = ∅