
Explore basic set operations such as union, intersection, complement, and set difference, including disjointness and universal set examples with even and odd natural numbers.
Explore the cartesian product, or cross product, as forming ordered pairs from two sets. See that A cross B has m into n elements and is not commutative.
Define a relation as any subset of the Cartesian product; on A, a relation is a subset of A x A, including the empty set.
Determine the number of relations on a set A with n elements by viewing A×A as n^2 ordered pairs; each pair is either included or not, giving 2^(n^2) relations.
Define a reflexive relation as containing every diagonal pair x,x for all x in A, illustrated by x-x, y-y, z-z in examples.
Explain that a subset of a reflexive relation need not be reflexive. Demonstrate that every superset, intersection, and union of reflexive relations is reflexive.
Explore the subset relation on a collection of sets and show it is reflexive because every set is a subset of itself.
Examine how to test symmetry in relations by analyzing parallel and perpendicular lines, less than and subset relations, and the 'is son of' relation, identifying which are symmetric.
Prove by contradiction that the intersection of two symmetric relations R1 and R2 is always symmetric, analyzing cases where b comma a appears or not in R1 and R2.
Show that every subset of a symmetric relation need not be symmetric, using R1 = {(a, b), (b, a)} on A = {a, b}, and supersets can fail symmetry.
Maximize the size by including all diagonal elements and, for each pair of distinct elements, include exactly one of (a,b) or (b,a); the maximum cardinality is n(n+1)/2.
Define an asymmetric relation as: if a relates to b, then b does not relate to a, and no diagonal elements are allowed; it is stricter than anti-symmetric.
Count all asymmetric relations on a set with n elements by excluding diagonal pairs and considering each unordered pair offers three choices, yielding 3^{(n^2 - n)/2} possibilities.
Explore the relationship between asymmetric and reflexive relations using Venn diagrams, illustrate with examples on a set A, and show that a relation cannot be both reflexive and asymmetric.
Explore the relationship between asymmetric and symmetric relations using a Venn diagram and A×A examples, showing the null relation is the only case that is both.
Identify the minimum and maximum cardinalities of transitive relations by examining the empty set and the full relation a×a, then analyze a sample R1 to show when transitivity fails.
Examine three relations on A = {1, 2, 3} to determine if they are partially ordered by testing reflexive, anti-symmetric, and transitive properties; none satisfy all conditions.
Show that the intersection of two equivalence relations is an equivalence relation by proving reflexivity, symmetry, and transitivity, using contradiction arguments.
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