Question

In: Math

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.

Expert Solution

Let be a finite set with elements then

A relation on the set is equivalnce if it satisfies

1) Reflexive Property :

2) Symmetric Property : If then

3) Transitive Property : and then

Each element of the set has an equivalence class

The equivalence class an element is the set of all those elements of the set with which the element has a relation .

The equivalence class of an element is denoted by .

The equivalence relation has the smallest number of equivalence classes .

The smallest equivalence relation must always contain the diagonal that is

Since every element is related to itself therefor it contains equivalence classes.

Since the set is non empty it contains elements. So, the smallest number of equivalence classes is equal to .

The largest number of equivalence classes in the set which contains elements is equal to because the largest relation in a set is the cartition product that is it contains elements.

For example the set with the relation .

The smallest ordered set is which has three elements that is the smallest number of equivalnce classes.

The largest ordered set of relation is

Which has elements which is the largest number of equivalence classes.

milcah answered 1 year 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 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.

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising...

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation. (c) (x1,y1)R(x2,y2) in R×R if x1∗y2 = x2∗y1.

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.

Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈ Z,b...

Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈ Z,b ∈ N} (recall that [(a,b)] = {(c,d) ∈Z×N : (a,b) ∼ (c,d)}). We deﬁne an addition and a multiplication on X as follows: [(a,b)] + [(c,d)] = [(ad + bc,bd)] and [(a,b)]·[(c,d)] = [(ac,bd)] Prove that this addition and multiplication is well-deﬁned on X.

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!

An equivalence relation partitions the plane R2 into the set of lines with slope 2. Describe...

An equivalence relation partitions the plane R2 into the set of lines with slope 2. Describe the relation on R .

Let f(n,k) be the number of equivalence relations with k classes on set with n elements....

Let f(n,k) be the number of equivalence relations with k classes on set with n elements. a) What is f(2,4)? b) what is f(4,2)? c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k * f(n-1,k)

Let (G,·) be a finite group, and let S be a set with the same cardinality...

Let (G,·) be a finite group, and let S be a set with the same cardinality as G. Then there is a bijection μ:S→G . We can give a group structure to S by defining a binary operation *on S, as follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) = g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}. First prove that (S,*) is a group. Then, what can you say about the bijection μ?

Let G be a simple graph. Prove that the connection relation in G is an equivalence...

Let G be a simple graph. Prove that the connection relation in G is an equivalence relation on V (G)