Discrete Math Series : Set Theory & Algebra masterclass

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.

Minimum and maximum cardinalities of a reflexive relation on a set with n elements: the smallest is the diagonal with n elements, the largest is A×A with n^2 elements.

Determine the number of reflexive relations on a set with n elements as 2^(n^2 − n), since all diagonal pairs must appear and off-diagonal pairs are optional.

Explain that a subset of a reflexive relation need not be reflexive. Demonstrate that every superset, intersection, and union of reflexive relations is reflexive.

Less than or equal to on real numbers is reflexive because every diagonal pair is included. The divisor relation is reflexive on non-zero real numbers.

Explore the subset relation on a collection of sets and show it is reflexive because every set is a subset of itself.

Define the parallel relation on the set of all lines. Show that it is reflexive because every line is parallel to itself.

Explain why proper subset relation on sets is not reflexive: a set is a subset of itself but lacks an extra element to qualify as proper, illustrated by diagonal elements.

Identify irreflexive relations on set A by excluding diagonal elements from R ⊆ A×A, contrasting with reflexive relations, and illustrate with A = {1,2} where 1 is related to 2.

Analyze how reflexive and irreflexive relations on set A are determined by diagonal elements, using R1, R2, and R3 as examples.

for a set with n elements, the smallest irreflexive relation is the empty set (cardinality 0); the largest is all non-diagonal pairs from A×A, with cardinality n^2−n.

Determine the number of irreflexive relations on an n-element set: exclude diagonal elements, leaving n^2 - n elements with two options each, giving 2^(n^2 - n) relations.

Explore reflexive and irreflexive relations on an n-element set. Use a Venn diagram to identify the neither region within the total 2^(n^2) relations.

Explore irreflexive relations through examples: less than on real numbers, proper subset on all sets, and subset, parallel, perpendicular relations on lines, noting diagonal elements.

Explore closure properties of irreflexive relations, showing subsets remain irreflexive; supersets may gain diagonal elements, while unions and intersections of irreflexive relations stay irreflexive.

Evaluate why the difference of two reflexive relations is not necessarily reflexive, showing that R1 − R2 may lack diagonal pairs and can be irreflexive through a proof-by-analysis approach.

Explore symmetric relations: if x is related to y, then y is related to x for all x,y in A. Use examples with A = {a,b,c} and A×A.

Explore the minimum and maximum cardinalities of symmetric relations on A = {1,...,n}, noting the empty relation has cardinality 0 and A×A has cardinality n^2.

The lecture introduces diagonal elements and jumbled ordered pairs in A cross A, explains that off-diagonal elements form combinations, and derives the number of symmetric relations as 2^(2^n) times 2^((n^2−n)/2).

Explore reflexive and symmetric relations on sets in the discrete math series masterclass, analyze R1 to R3, and visualize four relation categories with a Venn diagram.

On an n-element set, the number of relations that are both reflexive and symmetric equals 2^{(n^2 - n)/2}, since diagonals are fixed and off-diagonal pairs are either both present or both absent.

Count relations that are either symmetric or reflexive, showing reflexive relations equal 2 power n square - n divided by 2, then apply inclusion-exclusion to avoid double counting.

Learn how to count relations on an n-element set by category: neither reflexive nor symmetric; symmetric but not reflexive; reflexive but not symmetric, using 2^{n^2} total relations.

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.

The lecture shows that if R1 and R2 are symmetric relations, their union R1 ∪ R2 is always 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.

Examine whether R1 minus R2 is symmetric when both R1 and R2 are symmetric, and demonstrate that R1 - R2 remains symmetric under this condition.

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.

This lecture analyzes how relations on A = {1,2} can be symmetric, irreflexive, both, or neither, using R1-R4 and diagonal elements.

Analyze how symmetric and irreflexive relations relate via a venn diagram and diagonal exclusion. Count these dual-relations as 2^{(n^2 - n)/2} for a set with n elements.

Examine the relationship between symmetric and irreflexive relations using a venn diagram, count the regions, and derive totals from 2^{n^2} relations on an n-element set.

Explain anti-symmetric relations: if a,b are related, then b,a should not occur unless a=b; diagonal elements may appear, and examples show when a,b and b,a both present, it's not anti-symmetric.

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.

Count antisymmetric relations on an n-element set by selecting diagonal presence and, for each off-diagonal pair a,b and b,a, one of three options. The total is 2^n times 3^((n^2 - n)/2).

Explore relationship between symmetric and anti-symmetric relations with a Venn diagram, showing cases of both, symmetric but not anti-symmetric, anti-symmetric but not symmetric, and neither, from A with n elements.

examine how many relations are symmetric, antisymmetric, or both, showing that to be both requires omitting all off-diagonal pairs while freely choosing diagonal entries, yielding 2^n relations.

Explore reflexive and anti-symmetric relations with four-case examples, and show that on an n-element set the number of both reflexive and anti-symmetric relations equals 3^((n^2-n)/2).

Discover the difference between irreflexive and anti-symmetric relations on a set, with examples and a formula for counting relations that satisfy both: 3^{(n^2 - n)/2}.

Explain how antisymmetric relations behave under subsets and supersets, showing that subsets remain antisymmetric while supersets may break antisymmetry.

Analyze the closure properties of anti-symmetric relations: union can fail to be anti-symmetric, and a counterexample demonstrates this. Show that the intersection of anti-symmetric relations remains anti-symmetric.

Prove that the difference of two antisymmetric relations, R1 minus R2, is antisymmetric, and examine whether the complement of an antisymmetric relation must be antisymmetric, using counterexamples.

Examine anti-symmetric relations through examples like less than or equal to, is divisor of, subset and superset, and explain why parallel or perpendicular relations may fail to be anti-symmetric.

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.

Determine the maximum cardinality of an asymmetric relation on a set with n elements. Exclude diagonal pairs and pick one of each reciprocal pair, yielding (n^2 - n)/2.

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 counting relations that are either asymmetric or reflexive using a Venn diagram, noting their intersection is zero and the total equals the sum of the two counts.

Explore the relationship between asymmetric and irreflexive relations, with examples and a Venn diagram, and count relations using 2^(n^2) while subtracting symmetric relations to isolate irreflexive but not 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.

Explore the relationship between asymmetric and anti-symmetric relations, illustrated by a three-region Venn diagram, with examples of each case and methods to count such relations on an n-element set.

Explore the closure properties of asymmetric relations by testing whether subsets, supersets, unions, intersections, differences, and complements preserve asymmetry.

Explore transitive relations on sets, using 1-2 and 2-3 to justify 1-3, and examine how missing x,z or absence in A×A affects transitivity.

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.

Analyze transitivity of less than on real numbers, greater than or equal to, subset, proper subset, and divides, using xRy and yRz examples to show x is related to z.

Explore equivalence relations by examining reflexive, symmetric, and transitive properties with concrete examples on set A = {1, 2, 3}, and learn about smallest and largest equivalence relations by cardinality.

Explore equivalence relations on the real numbers by examining R1, where a equals b, and verify reflexive, symmetric, and transitive properties for a robust set-theory concept.

The lecture analyzes three relations: R2 not reflexive, so not equivalence; R3 reflexive but not symmetric, so not equivalence; R4 not reflexive, so not equivalence.

R5, defined by a minus b is even, is reflexive, symmetric, and transitive, hence an equivalence relation.

Define the partially ordered relation on a set as reflexive, anti-symmetric and transitive, and distinguish anti-symmetric from asymmetric while examining examples.

Identify the only relation on A that is both an equivalence relation and a partial order: the identity relation with all diagonal elements and no off-diagonal pairs.

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.

Identify a po set as set A with a partially ordered relation R on A, written as (A, R). Verify the relation is partial by being reflexive, antisymmetric, and transitive.

Demonstrates why several relations form POSETs by verifying reflexive, anti-symmetric, and transitive properties. Covers examples including real numbers with ≤, subset, and divisibility relations as POSETs.

Analyzing a relation R on A = {a,b,c,d}, the lecture shows R is not reflexive and not anti-symmetric, but it is transitive, implying R is not an equivalence relation.

Analyzing a relation on a collection of sets defined by a ∩ b = π shows it is not reflexive, is symmetric, and not transitive.

If R1 and R2 are equivalence relations, their union is reflexive and symmetric, but may not be transitive, so R1 union R2 need not be an equivalence relation.

Show that the intersection of two equivalence relations is an equivalence relation by proving reflexivity, symmetry, and transitivity, using contradiction arguments.

Analyze whether given relations on integers form an equivalence relation by testing reflexivity, symmetry, and transitivity. R1 fails reflexivity, while R2 is reflexive, symmetric, and transitive.

This lecture defines a relation on natural numbers using |a-b| ≤ 2 and shows it is reflexive and symmetric but not transitive, hence not an equivalence relation.

Analyze the relation R4: a*b > 0 on real numbers; show it is not reflexive for all reals, but is reflexive, symmetric, and transitive on nonzero real numbers.

Analyze the symmetry, anti-symmetry, and transitivity of the relations R1 through R4 on A = {a, b, c}, and identify that the second option is false.

This lecture analyzes a relation on ordered pairs of integers defined by x<u and y<v, showing it is not reflexive, thus not a partial order or an equivalence relation.

Understand the totally ordered set (TOSET) as a POSET where every pair is comparable. See real numbers with ≤ and the divisibility relation on {2,4,6}, illustrating when comparability holds.

Examine the divides relation as a poset, verify reflexive, antisymmetric, and transitive properties, and show {1,2,3,4,5,6,7} is not a totally ordered set, unlike the totally ordered set {1,2,6,30,60,300}.

Explore how a poset is described by a Hasse diagram, showing a reflexive, anti-symmetric, and transitive relation with vertical lines and implicit transitivity.