1. Consider the relations R = {(x,y),(y,z),(z,x)} and S =
{(y,x),(z,y),(x,z)} on {x, y, z}. a)...
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?
Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x,...
Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x, y) ∈ R if and only if x + 2 > y.
For example, (4, 3) is in R because 4 + 2 = 6, which is greater
than 3.
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.
For each of the following, prove that the relation is an
equivalence relation. Then give the...
For each of the following, prove that the relation is an
equivalence relation. Then give the information about the
equivalence classes, as specified.
a) The relation ∼ on R defined by x ∼ y iff x = y or xy = 2.
Explicitly find the equivalence classes [2], [3], [−4/5 ], and
[0]
b) The relation ∼ on R+ × R+ defined by (x, y) ∼ (u, v) iff x2v
= u2y. Explicitly find the equivalence classes [(5, 2)] and...
Determine whether the given relation is an equivalence relation
on {1,2,3,4,5}. If the relation is an...
Determine whether the given relation is an equivalence relation
on {1,2,3,4,5}. If the relation is an equivalence relation, list
the equivalence classes (x, y E {1, 2, 3, 4, 5}.)
{(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1), (3,4),
(4,3)}
If the relation above is not an equivalence relation, state that
the relation is not an equivalence relation and why.
Example: "Not an equivalence relation. Relation is not
symmetric"
Remember to test all pairs in relation R
Problem 13.5. Consider a “square” S = {(x, y) : x, y ∈ {−2, −1,
0,...
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...
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)
Consider the relation R defined on the real line R, and defined
as follows: x ∼...
Consider the relation R defined on the real line R, and defined
as follows: x ∼ y if and only if the distance from the point x to
the point y is less than 3. Study if this relation is reflexive,
symmetric, and transitive. Which points are related to 2?