Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only...
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...
[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.
Define a relation R on N x N by R = {(x,y) | x ε N, y ε N and x+y is even} Prove or disprove: R is an equivalence relation.
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
Define a relation from R to R by saying that (x,y) ES if and only if 3x² + y2 = 25 (a) List five different elements of S. (b) Prove that S is not a function.
Define an equivalence relation on R by (x,y,z) ∼ (u,v,w) whenever x +y +z = u +v +w . Describe the equivalence classes.
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)...
8.) Consider the integers Z. Dene the relation on Z by x y if and only if 7j(y + 6x). Prove: a.) The relation is an equivalence relation. b.) Find the equivalence class of 0 and prove that it is a subgroup of Z with the usual addition operator on the integers. 8.) Consider the integers Z. Define the relation ~ on Z by x ~ y if and only if 7)(y + 6x). Prove: a.) The relation is an...
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...
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.