4. Prove that {(x,y) e R2 : x - ye Q} is an equivalence relation on...
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...
problem 23 please :) and here is Q.21 Problem 23. Recall from Problem 21 the equivalence relation ~ on the set of rational Cauchy sequences C. Define 〈z) E C to be eventually positive if there is an M є N such that xn > 0 for all Prove that eventually positive is a well defined notion on c/ (z〉 ~ 〈y), then 〈y〉 İs eventually positive. ie. if 〈z) is eventually positive and Problem 21. Let C be the...
Prove that the following relation R is an equivalence relation on the set of ordered pairs of real numbers. Describe the equivalence classes of R. (x, y)R(w, z) y-x2 = z-w2
Define the set F- (XI X is a finite set of counting numbers) and the relation is a finiice sei of counting nuobors and the relation {(X Z〉 | Ye F and Z € Fand y-2). This relation is just a version of the usual subset relation, but restricted to only apply to the sets in F Prove: CFis a partial order. Prove: Cis not symmetric and connected. Prove: If R is an equivalence relation, it is also a euclidean...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
Please do problem 9 and write a detailed proof when doing (a) 9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b) Determine the equivalence class of π. 9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b)...
Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3] Z). A relation R is defined on the set Q+ of positive rational numbers by R b if Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3] Z). A relation R is defined on the set Q+ of positive rational numbers by R...
Determine if {(x,y) | x divides 2-y} is an equivalence relation on {1,2,3,4,5}. List the equivalence classes Determine if {(x,y) | x and y are both even or x and y are both odd} is an equivalence relation on {1,2,3,4,5}. List the equivalence classes. Determine if {(x,y) | x and y are the same height} is an equivalence relation on all people Determine if {(x,y) | x and y have the same color hair} is an equivalence relation on all...
Prove that if R is an equivalence relation on a set A, then R ^-1 is an equivalence relation on A.
2. (5pt) Consider the following binary relations. In each case prove the relation in question is an equivalence relation and describe, in geometric terms, what the equivalence classes are. (a) Si is a binary relation on R2 x R2 defined by z+ly-+ 1 r,y). (,y) e S Recall that R =R x R. (b) Sa is a binary relation on R defined by 1-ye2 r,y) e S