Suppose R is the relation defined on all real numbers by for all real numbers x,y...
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.
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?
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...
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)...
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...
A is the binary relation defined on real numbers as follows. For all real numbers 1, 39 XAy if and only if xy >0. Determine if A is reflexive, symmetric, transitive, antia symmetric.
Consider the relation R on the real numbers where xRy if and only if xy = 1. (a) What is R^2 ? (b) What is R^3 ? (c) What is R^i for i ≥ 1? (d) What is R^∗ ? I really don't understand the concept. can you explain it with details?
Suppose that the function f is defined, for all real numbers, as follows. f(x) = x-2 ifx#2 4 if x=2 Find f(-3), f(2), and f(5). s(-3) = 0 s(2) = 0 r(s) = 1 Suppose that the function g is defined, for all real numbers, as follows. if x -2 8(x)= 1-4 if x=-2 Find g(-5), g(-2), and g(4). $(-5) = 0 DO s(-2) = 1 8(4) = 1
Suppose that the functions s and t are defined for all real numbers x as follows. (r)-5r t(r) 3x-2 Write the expressions for (+s)(x) and (-)) and evaluate (t s)X-2). +3)-0 ts(x) s)(x) = (rs)(-2)D
Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x + y = 0 b) x= ±y. c) x-y is a rational number. d) = 2y. e) xy ≥ 0. f) xy = 0. g) x=l. h) r=1 or y = 1