Let x be a set and let R be a relation on x such x is...

Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

Expert Solution

Colby Messinger answered 2 years ago

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that X / R is a partition of X.

Let R be the relation on the set of people given by aRb if a and...

Let R be the relation on the set of people given by aRb if a and b have at least one parent in common. Is R an equivalence relation? (Equivalence Relations and Partitions)

Let R be a relation on a set that is reflexive and symmetric but not transitive?...

Let R be a relation on a set that is reflexive and symmetric but not transitive? Let R(x) = {y : x R y}. [Note that R(x) is the same as x / R except that R is not an equivalence relation in this case.] Does the set A = {R(x) : x ∈ A} always/sometimes/never form a partition of A? Prove that your answer is correct. Do not prove by examples.

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

let G be a simple graph. show that the relation R on the set of vertices...

let G be a simple graph. show that the relation R on the set of vertices of G such that URV if and only if there is an edge associated with (u,v) is a symmetric irreflexive relation on G

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring...

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring of the functions from X to R. Show directly the associativity of the multiplication of F(X, R). Assume that R is unital and commutative. show that F(X, R) is also unital and commutative.

Let V be the set of positive reals, V = {x ∈ R : x >...

Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is...

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.

Assume A = R and the relation R ⊆ A × A such that for x,...

Assume A = R and the relation R ⊆ A × A such that for x, y, ∈ R, xRy if and only if sin2 x + cos2 y = 1. Prove that R is an equivalence relation and for any fixed x ∈ R, find the equivalence class x

Let X be the set of all subsets of R whose complement is a finite set...

Let X be the set of all subsets of R whose complement is a finite set in R: X = {O ⊂ R | R − O is finite} ∪ {∅} a) Show that T is a topological structure no R. b) Prove that (R, X) is connected. c) Prove that (R, X) is compact.

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