Sets and Relations in Mathematics

Sets

• Mathematics serves as the foundational language for theoretical computation, emphasizing the importance of sets.
• A set is defined as a collection of distinct objects, known as elements or members.
• Example: Set L can be represented as L = {a, b, c, d}.
• Elements can be expressed using the notation ( b \in L ) (b is in L) or ( z \notin L ) (z is not in L).
• Sets do not consider element repetitions or order; {red, blue, red} is equivalent to {red, blue}.
• Two sets are equal only if they contain the exact same elements.
• Elements of sets can be varied and unrelated, exemplified by {3, red, {d, blue}}.
• A singleton set contains only one element, e.g., {1}.
• The empty set, denoted by ∅ or {}, is the unique set with no elements, while any non-empty set is termed nonempty.
• Finite sets can be fully listed, while infinite sets cannot, e.g., the set of natural numbers, represented as ( N = {0, 1, 2, \ldots} ).

Set Definitions and Subsets

• Sets can be defined through properties of their elements; e.g., if ( I = {1, 3, 9} ), the subset ( G ) can be defined as ( G = {x : x \in I \text{ and } x > 2} ).
• Subset notation ( A \subseteq B ) indicates all elements of A are in B, while ( A \subset B ) represents a proper subset (A is not equal to B).
• The empty set is a subset of every set: ( \emptyset \subseteq B ).
• To demonstrate that two sets are equal, prove both ( A \subseteq B ) and ( B \subseteq A ).

Set Operations

• Union of sets combines all unique elements from both, expressed as ( A \cup B = { x : x \in A \text{ or } x \in B } ).
• Example: ( {1, 3, 9} \cup {3, 5, 7} = {1, 3, 5, 7, 9} ).
• Intersection includes elements common to both sets, denoted as ( A \cap B = { x : x \in A \text{ and } x \in B } ).
• Example: ( {1, 3, 9} \cap {3, 5, 7} = {3} ).
• Set difference ( A - B ) consists of elements in A that aren't in B, expressed as ( A - B = { x : x \in A \text{ and } x \notin B } ).
• Example: ( {1, 3, 9} - {3, 5, 7} = {1, 9} ).
• Properties of set operations include Idempotency laws:
  • ( A \cup A = A )
  • ( A \cap A = A )