Let H ⊂ R2 be the line H = {(a, 2a) : a ∈ R}. Consider the collection of all translates of H, i.e., all lines in the plane with slope 2. Find the equivalence relation on R2 defined by this partition of R2 .
Let H ⊂ R2 be the line H = {(a, 2a) : a ∈ R}. Consider the collection of all translates of H, i.e...
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2 {(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
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...
proofs there isnt anymore info for the question Exercise 5.3.9. For each of the following equivalence relations, describe the corre- sponding partition. Your description of each partition should have no redundancy, and should not refer to the name of the relation. (1) Let P be the set of all people, and let be the relation on P defined by x y if and only if x and y have the same mother, for all x,y e P. (2) Let ~...
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...
3. Let the relation R be defined on the set R by a Rb if a -b is an integer. Is R and equivalence relation? If yes, provide a proof. Consider the equivalence relation in #3. a. What is the equivalence class of 3 for this relation? 1 b. What is the equivalence class of for this relation? 2
For r ∈R, let Ar = {(x,y,z ∈R^3 . . . x^2 + y^2 −z^2 = r}. Is this a partition of R^3? If so, give a geometric description of the partitioning sets (i.e., the equivalence classes of the induced equivalence relation).
Please answer all!! 17. (a) Let R be the relation on Z be defined by a R b if a² + 1 = 62 + 1 for a, b e Z. Show that R is an equivalence relation. (b) Find these equivalence classes: [0], [2], and [7]. 8. Let A, B, C and D be sets. Prove that (A x B) U (C x D) C (AUC) Ⓡ (BUD).
*ESPECIALLY PART D PLEASE 111111 1. Let R be a relation on RxR defined by (a,b)R(c,d) if and only if a - b = c-d DIDUD a) (5 points) Prove that is an equivalence relation on RxR. b) (5 points) Describe all ordered pairs in the equivalence class of (0,0) c) (5 points) Describe all ordered pairs in the equivalence class of (3,1) d) (5 points) Describe the partition of Rx Rassociated with R.
QI. Let A-(-4-3-2-1,0,1,2,3,4]. R İs defined on A as follows: For all (m, n) E A, mRn㈠4](rn2_n2) Show that the relation R is an equivalence relation on the set A by drawing the graph of relation Find the distinct equivalence classes of R. Q2. Find examples of relations with the following properties a) Reflexive, but not symmetric and not transitive. b) Symmetric, but not reflexive and not transitive. c) Transitive, but not reflexive and not symmetric. d) Reflexive and symmetric,...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...