6. Let R be the relation defined on Z by a Rb if a + b is even. Show that Ris an equivalence relation.
Question 8 Let R be relation on a set A. 1. When is R said to be an equivalence relation? Give a precise definition, using appropriate quantifiers etc. 2. When is R said to be an partial order? Give a precise definition, using appropriate quantifiers etc (You don't need to redefine things that you defined in the previous part... you may simply mention them to save time.) 3. On Z, define a relation: a D biff a - b is...
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...
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) Element R if and only if ad = bc. Show that R is an equivalence relation What is the equivalence class of of (1, 2), i.e. [(1, 2)]?
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)...
(4) Let R be the relation on N given by a Rb if al5b or bl5a. Decide if it is an equivalence relation.
Let A = ( a, b, c, d ) and let ( A, R ) be a posset where R is a Relation on A defined by: R is reflexive c ≤ d a ≤ c a ≤ b a ≤ d b ≤ d Find H(A) Is (A, R) a lattice? If you answer no, give a counterexample. If you answer yes, give a brief justification as to why (no formal proof needed). Is (A,R) a Boolean algebra? Give...
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...
(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 :=...
3. (a) Let R be a binary relation on the set X = {1,2,3,4,5,6,7}, defined by R= {(1,3), (2,3), (3, 4), (4,4),(4,5), (5,6), (5,7)} (1) (6 pts) Find Rk for all k = 2, 3, 4, 5,... (2) (3 pts) Find the transitive closure t(R) of R by Washall's algorithm and draw the directed graph of t(R).