The ordered pairs in the partition consists of all possible pairs from each class.
So, P = { (a,c) , (a,e), (c,a), (c.e), (e,a), (e,c), (b,f), (f,b), (d,d) }
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by ...
Complete the proof of Theorem 4.22 by showing that < is a transitive relation. Let R be a transitive relation that is reflexive on a set S, and let E-ROR-1. Then E is an equivalence relation on S, and if for any two equivalence classes [a] and [b] we define [a] < [b] provided that for each x e [a] and each y e [b], (x, y) e R, then (S/E, is a partially ordered set.
The assertion, as given in the source: (Theorem 3.4.8 of Johnsonbaugh's text.) Let R be an equivalence relation on a set X. For each a in X, define [a] as (xeX | xRaj. Then the following set is a partition of X: S={[a] l a eX). Logical structure of the assertion: Proof framework, based on this logical structure: Previous resulis needed (give one to three previous resulis that are most important for this prooj): Interesting or unexpected tricks, or summary:...
And Heres theorem 10.1 Prove that the relation VR of Theorem 10,1 is an equivalence relation. ① show that a group with at least two elements but with no proper nontrivite subgroups must be finite and of prime order. 10.1 Theorem Let H be a subgroup of G. Let the relation ~1 be defined on G by a~lb if and only if albe H. Let ~R be defined by a~rb if and only if ab- € H. Then ~1 and...
Problem 11.16. Let X = {XE Ζ+ : x-100): that is, X is the set of all integers from l to 100. For each Y E 9(X) we define AY (2 E 9(X) : Y and Z have the same number of elements) (a) Prove that AY : Y є 9(X)} partitions 9(X). (b) Letdenote the equivalence relation on (X) that is associated with this partition (according to Theorem 11.4). If possible, find A, B, and C such that 1....
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y be a function with domain as the quotient set X/ ∼ and codomain as some set Y . We define a function ˜f, called the lift of f, as follows: ˜f : X → Y, x 7→ f([x]). We define a function Φ : F(X/ ∼, Y ) → F(X, Y ), f 7→ ˜f. (1) Is Φ injective? Give a proof or a...
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...