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.

Expert Solution

Colby Messinger answered 2 years ago

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 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.

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R...

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R is an equivalence relation, that is, prove that it is reflexive, symmetric, and transitive. Determine the equivalence classes of this relation. What members are in the class [2]? How many members do the equivalence classes have? Do they all have the same number of members? How many equivalence classes are there? Question 2. Equivalence Relation 2 Consider the relation from last week defined as:...

Let R and S be equivalence relations on a set X. Which of the following are...

Let R and S be equivalence relations on a set X. Which of the following are necessarily equivalence relations? (1)R ∩ S (2)R \ S . Please show me the proof. Thanks!

Let X be a finite set. Describe the equivalence relation having the greatest number of distinct...

Let X be a finite set. Describe the equivalence relation having the greatest number of distinct equivalence classes, and the one with the smallest number of equivalence classes.

Let X be a non-empty set and P(X) its power set. Then (P(x), symetric difference, intersection)...

Let X be a non-empty set and P(X) its power set. Then (P(x), symetric difference, intersection) is a ring. Find a non-trivial ideal of P(X).

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.

Question

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

Solutions

Expert Solution

Related Solutions