9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff b|d and alc (a) R is symmetric but not reflexive. (b) R is transitive and symmetric but not reflexive (c) R is reflexive and transitive but not symmetric (d) None of the above 10. Let R be an equivalence relation on a nonempty and finite 9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff b|d and alc...
(i) Prove that the realtion in Z of congruence modulo p is an equivalence relation. Namesly, show that Rp := {(a,b) € ZxZ:a = 5(p)} is reflexive, symmetric and transitive. (ii) Let pe N be fixed. Show that there are exactly p equivalence classes induced by Rp. (iii) Consider the relation S E N defined as: a Sb if and only if a b( i.e., a divides b). Prove that S is an order relation. In other words, S :=...
1. Define a relation on Z by aRb provided a -b a. Prove that this relation is an equivalence relation. b. Describe the equivalence classes. 2. Define a relation on Z by akb provided ab is even. Use counterexamples to show that the reflexive and transitive properties are not satisfied 3. Explain why the relation R on the set S-23,4 defined by R - 11.1),(22),3,3),4.4),2,3),(32),(2.4),(4,2)) is not an equivalence relation.
4. Consider the relation on the positive integers xRy if and only if x x+y (a) List three ordered pairs from this relationship (b) Is R reflexive? Prove your answer (c) Is R symmetric? Prove your answer (d) Is R anti-symmetric? Prove your answer (e) Is R transitive? Prove your answer.
Let S be the set of all subsets of Z. Define a relation,∼, on S by “two subsets A and B of Z are equivalent,A∼B, if A⊆B.” Prove or disprove each of the following statements: (a)∼is reflexive(b)∼is symmetric(c)∼is transitive
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,...
. Define a relation ∼ on R 2 by stating that (a, b) ∼ (c, d) if and only if a 2 + b 2 ≤ c 2 + d 2 . Show that ∼ is reflexive and transitive but not symmetric.
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the relation is reflexive (c) (6pt) Prove that the relation is not symmetric. (d) (6pt) Prove that the relation is transitive. (17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f...
10. Represent the following relation R on the set (. t,s t) as a digraph and as a acro-ooe ati eachi b) Circle the properties of R antisymmetric itive 2 peins) reflexive symmetric 떼 1 find the symmetric and transitive closures ofS (4 pirtepi 1 c)If Ms-10 Transitive Closure Symmetric Closure_-
3. (a) If aRb is a relation of congruent modulo n, a ≡ b (mod n). Show that R is: (i) reflexive. (ii) symmetric. (iii) transitive. (b) A is a set and | A | = 8. R is a relation on A, R ⊆ A X A. (i) How many different R can be produced? (ii) How many R are reflexive? (iii) How many R are symmetric? (iv) How many R are reflexive and symmetric? (c) A computer application...