home / study / math / other math / questions and answers / 1. let s be a nonempt

Question

home / study / math / other math / questions and answers / 1. let s be a nonempty set and let b be a fixed ...

Your question has been answered

Let us know if you got a helpful answer. Rate this answer

Question: 1. Let S be a nonempty set and let B be a fixed su...

Bookmark

1. Let S be a nonempty set and let B be a fixed subset of S. A relation R defined on the set of subsets of S is defined by X R Y if X B = Y B. (a) Prove that R is an equivalence relation. (b) Let S = {1, 2, 3, 4, 5} and B = {1, 3, 5}. For X = {1, 2, 3}, find [X].

2. Let f : A B and let C1, C2 A. (a) Prove that f(C1) f(C2) f(C1 C2). (b) Find an example of f, C1 and C2 where equality doesn’t hold. (c) Prove that equality does hold if f is bijective.

3. Let f : A B, g : B C, and h : B C be functions where f is bijective. (a) Prove that if g f = h f then g = h. (b) Show that part (a) is false if f is not bijective.

4. Let S be a set. Prove that bd S S 0 iso S where iso S denotes the set of isolated points of S.

5. Let S and T be sets.

(a) Prove that int S int T int (S T).

(b) Give an example of sets S and T where equality does not hold.

6. For each of the following, either prove or give a counterexample:

(a) Let A, B, C, and D be sets with A C and B D. If A and B are disjoint then C and D are disjoint.

(b) Let A and B be sets. If A B = B A, then A B = . (c) Let R1 and R2 be two equivalence relations on a nonempty set A. Then R1 R2 is an equivalence relation on A.

Explanation / Answer

Ans-

A relation on a nonempty set SS that is reflexive, symmetric, and transitive is said to be an equivalence relation on SS. Thus, for all x,y,zSx,y,zS,

As the name and notation suggest, an equivalence relation is intended to define a type of equivalence among the elements of SS. Like partial orders, equivalence relations occur naturally in most areas of mathematics, including probability.

Suppose that is an equivalence relation on SS. The equivalence class of an element xSxS is the set of all elements that are equivalent to xx, and is denoted[x]={yS:yx}[x]={yS:yx}

Results

The most important result is that an equivalence relation on a set SS defines a partition of SS, by means of the equivalence classes.

Suppose that is an equivalence relation on a set SS.

Proof:

Sometimes the set SS of equivalence classes is denoted S/S/. The idea is that the equivalence classes are new objects obtained by identifying elements in SS that are equivalent. Conversely, every partition of a set defines an equivalence relation on the set.

Suppose that SS is a collection of nonempty sets that partition a given set SS. Define the relation on SS by xyxy if and only if xAxA and yAyA for some ASAS.

An equivalence relation on a non empty set can't be empty, because it's reflexive. So, for any aAaA, you have (a,a)R(a,a)R. Now there is some aAaA.

From a slightly different point of view, an equivalence relation on AA always contains the identity relation

A={(a,a):aA}A={(a,a):aA}

It's true that the empty relation is transitive and symmetric (also antisymmetric, by the way) on every set.

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.