Consider the relation Rover the set N x N:(mi, ni)R(m2, 12) if mi + mzis even...
(14) Let R be a relation on the integers defined by m R n if and only if m+m2 n+ n2(mod 5). Show that R is an equivalence relation and determine all the equivalence classes.
Prove that the following relation R is an equivalence
relation on the set of ordered pairs of real numbers. Describe the
equivalence classes of R. (x, y)R(w, z)
y-x2 = z-w2
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...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
Define a relation R on N x N by R = {(x,y) | x ε N, y ε N and x+y is even} Prove or disprove: R is an equivalence relation.
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...
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.
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...
Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of f(x) ??
Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of...
(10) Define a relation R on Zn (the integers mod n) as follows: lal isR related to [b (i.e. [an Rbn) iff there is [cn E Gn such that a b (a) Show that R is an equivalence relation on Zn (b) Give all the equivalence classes for R when n-12.