Discrete Mathematics.
Let A = {2,4,6,8,10}, and define a relation R on A as ∀x,y ∈ A,xRy ↔ 4|(x−y).
(a) Show R is an equivalence relation.
(b) Give R explicitly in terms of its elements.
(c) Draw the directed graph of R.
(d) List all the distinct equivalence classes of R.
c)
Discrete Mathematics. Let A = {2,4,6,8,10}, and define a relation R on A as ∀x,y ∈...
Discrete Mathematics. Let A = {2,3,4,6,8,9,12,18}, and define a relation R on A as ∀x,y ∈ A,xRy ↔ x|y. (a) Is R antisymmetric? Prove, or give a counterexample. (b) Draw the Hasse diagram for R. (c) Find the greatest, least, maximal, and minimal elements of R (if they exist). (d) Find a topological sorting for R that is different from the ≤ relation.
13 pts) Let R be the relation on R deÖned by xRy means "sin2 (x) + cos2 (y) = 1". Recall the Pythagorean identity: 8u 2 R we have sin2 (u) + cos2 (u) = 1. (a) (9 pts) PROVE that R is an equivalence relation on R. (b) (4 pts) Describe all elements of the (inÖnite) equivalence class [0]. Recall: sin(0) = 0 and cos(0) = 1. 2. (13 pts) Let R be the relation on R defined by...
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 R be the relation on N defined by xRy iff 2 divides x+y. R is an equivalence relation. You do not have to prove that R is an equivalence relation. True or False: 3 ∈ 4/R.
8. On the set A = {1,2,3,4,...,20}, an equivalence relation R is defined as follows: For all x, y € A, xRy 4(x - y). For each of the following, circle TRUE or FALSE. [4 points) a. TRUE or FALSE: There are only 4 distinct equivalence classes for this relation. b. TRUE or FALSE: If you remove all the even numbers from A, the relation would still be an equivalence relation. C. TRUE or FALSE: In this equivalence relation, 2R5...
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...
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...
Please how to solve this problem! thank you & (-76 Points) DETAILS EPPDISCMATHS 8.3.006. MY NOTES ASK YOUR TEACHER Let A = (-2,-1, 0, 1, 2, 3, 4, 5, 6, 7) and define a relation Ron A as follows: For all x,y EA, XRY 3)(x - y). It is a fact that is an equivalence relation on A. Use set-roster notation to write the equivalence classes of R. [0] = [1] - [2] - (3) - How many distinct equivalence...
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,...
Let X, be the set {x € Z|3 SXS 9} and relation M on Xz defined by: xMy – 31(x - y). (Note: Unless you are explaining “Why not,” explanations are not required.) a. Draw the directed graph of M. b. Is M reflexive? If not, why not? C. Is M symmetric? If not, why not? d. Is M antisymmetric? If not, why not? e. Is M transitive? If not, why not? f. Is M an equivalence relation, partial order...