Question

1. Consider the relations R = {(x,y),(y,z),(z,x)} and S = {(y,x),(z,y),(x,z)} on {x, y, z}. a) Explain why R is not an equivalence relation. b) Explain why S is not an equivalence relation. c) Find S ◦ R. d) Show that S ◦ R is an equivalence relation. e) What are the equivalence classes of S ◦ R?

Answer #1

Problem 57 on page 617 from
Rosen) Consider the equivalence relation R = {(x, y)| x-y is an
integer}
a. What is the equivalence
class of 1 for this equivalence relations?
b. What is the equivalence
class of 1/2 for this equivalence relation?

Consider the following relation on the set Z: xRy ?
x2 + y is even.
For each question below, if your answer is "yes", then prove it, if
your answer is "no", then show a counterexample.
(i) Is R reflexive?
(ii) Is R symmetric?
(iii) Is R antisymmetric?
(iv) Is R transitive?
(v) Is R an equivalence relation? If it is, then describe the
equivalence classes of R. How many equivalence classes are
there?

Consider the following set S = {(a,b)|a,b ∈ Z,b 6= 0} where Z
denotes the integers. Show that the relation (a,b)R(c,d) ↔ ad = bc
on S is an equivalence relation. Give the equivalence class
[(1,2)]. What can an equivalence class be associated with?

Prove the following: Theorem. Let R ⊆ X × Y and S ⊆ Y × Z be
relations. Then
1. Range(S ◦ R) ⊆ Range(S), and
2. if Domain(S) ⊆ Range(R), then Range(S ◦ R) = Range(S)

Let S be the set of all functions from Z to Z, and consider the
relation on S:
R = {(f,g) : f(0) + g(0) = 0}.
Determine whether R is (a) reﬂexive; (b) symmetric; (c)
transitive; (d) an equivalence relation.

Consider the mapping R^3 to R^3 T[x,y,z] = [x-2z, x+y-z, 2y]
a) Show that T is a linear Transformation
b) Find the Kernel of T
Note: Step by step please. Much appreciated.

Let
R
=
{(x, y) | x − y is an
integer}
be a relation on
the set Q of rational numbers. a)
[6
marks] Prove
that R is an equivalence relation
on Q.
b) [2
marks] What
is the equivalence class of 0?
c) [2
marks] What
is the equivalence class of 1/2?

Problem 13.5. Consider a “square” S = {(x, y) : x, y ∈ {−2, −1,
0, 1, 2}}. (a) Let (x, y) ∼ (x 0 , y0 ) iff |x| + |y| = |x 0 | + |y
0 |. It is an equivalence relation on S. (You don’t need to prove
it.) Write the elements of S/ ∼. (b) Let (x, y) ∼ (x 0 , y0 ) iff •
x and x 0 have the same sign (both...

Define a relation on N x N by (a, b)R(c, d) iff ad=bc
a. Show that R is an equivalence relation.
b. Find the equivalence class E(1, 2)

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

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