Question

Let R be the relation on the integers given by (a, b) ∈ R ⇐⇒ a − b is even.

1. Show that R is an equivalence relation

2. List teh equivalence classes for the relation

Can anyone help?

Answer #1

Consider the following relation ∼ on the set of integers
a ∼ b ⇐⇒ b 2 − a 2 is divisible by 3
Prove that this is an equivalence relation. List all equivalence
classes.

Let N* be the set of positive integers. The relation
∼ on N* is defined as follows: m ∼ n ⇐⇒ ∃k ∈
N* mn = k2
(a) Prove that ∼ is an equivalence relation.
(b) Find the equivalence classes of 2, 4, and 6.

Let R be the relation of congruence mod4 on Z: aRb if a-b= 4k,
for some k E Z.
(b) What integers are in the equivalence class of 31?
(c) How many distinct equivalence classes are there? What are
they?
Repeat the above for congruence mod 5.

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q.
I need to prove that:
a) R is an equivalence relation. (which I have)
b) The equivalence classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq
I...

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Show that the equivalence
classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq.
you may use the following lemma: If p is prime...

Let R be a relation on a set of integers, which is represented
by:
a R b if and only if a = 2 ^ k.b, for some integer k.
Check if the relation R is an equivalent relation!

Let R be the relation on Z defined by:
For any a, b ∈ Z , aRb if and only if 4 | (a + 3b). (a) Prove that
R is an equivalence relation.
(b) Prove that for all integers a and b, aRb if and only if a ≡
b (mod 4)

Let
A be the set of all integers, and let R be the relation "m divides
n." Determine whether or not the given relation R, on the set A, is
reflexive, symmetric, antisymmetric, or transitive.

Using Discrete Math
Let ρ be the relation on the set of natural numbers N given by:
for all x, y ∈ N, xρy if and only if x + y is even. Show that ρ is
an equivalence relation and determine the equivalence classes.

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 16 minutes ago

asked 30 minutes ago

asked 34 minutes ago

asked 40 minutes ago

asked 53 minutes ago

asked 56 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago