

4. Consider the relation on the positive integers xRy if and only if x x+y (a)...
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
4. Give the directed graph of a relation on the set ( x,y,z that is a) not reflexive, not symmetric, but transitive b) irreflexive, symmetric, and transitive c) neither reflexive, irreflexive, symmetric, antisymmetric, nor transitive d) a poset but not a total order e) a poset and a total order
4) Define a relation TC Nx N such that T = {(a,b) a EA A DEA 18- b = 2c+1 for some integer c}. (N is the set of non-negative integers.) a) Prove that this relation is not reflexive. b) Prove that this relation is symmetric. c) Define the term anti-transitive as the following: Given a set A and a relation R, if for all a,b,ceA, (aRb a bRc A cRa) = (a = b v b= c) Prove that...
Discrete Math
12. Consider the relation Con R given by xCy if and only if x - y < 10. (a). Determine if C is reflexive. (b). Determine if C is symmetric. (c). Determine if C is transitive.
10. [12 Points) Properties of relations Consider the relation R defined on R by «Ry x2 - y2 = x - y (a) Show that R is reflexive. (b) Show that R is symmetric. (c) Show that R is transitive. (d) You have thus verified that R is an equivalence relation. What is the equivalence class of 3? (e) More generally, what is the equivalence class of an element x? Use the listing method. (f) Instead of proving the three...
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...
(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 :=...
Can you #2 and #3?
6. LESSON 6 (1) Let A be the set of people alive on earth. For each relation defined below, determine if it is an equivalence relation on A. If it is, describe the equivalence classes. If it is not determine which properties of an equivalence relation fail. (a) a Hb a and b are the same age in (in years). (b) a Gb a and b have grandparent in common. 2) Consider the relation S(x,y):x...
For natural numbers x and y, define xRy if and only if x^2 + y is even. Prove that R is an equivalence relation on the set of natural numbers and find the quotient set determined by R. What would the quotient set be? can this proof be explained in detail?
Let the relation R be defined on the set {x ∈ R | 0 ≤ x ≤ 1} by xRy ⇔ ∃t(x + t = y and 0 ≤ t ≤ 1) Is R transitive?