Question

1. We define a relation C on the set of humans as xRy ⇐⇒ x and y were born in the same country

Describe the equivalence class containing yourself as an element.

2. Let R be an equivalence relation with (x, y) ∈ R and (y, z) is not ∈ R (that is, y does not relate to z). Can you determine whether or not xRz? Why or why not?

Answer #1

Prove that the relation R on the set of all people, defined by
xRy if x and y have the same first name is an 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?

For natural numbers x and y, define xRy if and only if
x^2 + y is even. Prove that R is an equivalence relation on the set
of natural numbers and find the quotient set determined by R. What
would the quotient set be? can this proof be explained in
detail?

Let R be the relation on the set of real numbers such that xRy
if and only if x and y are real numbers that differ by less than 1,
that is, |x − y| < 1. Which of the following pair or pairs can
be used as a counterexample to show this relation is not an
equivalence relation?
A) (1, 1)
B) (1, 1.8), (1.8, 3)
C) (1, 1), (3, 3)
D) (1, 1), (1, 1.5)

Let
A = {1,2,3,4,5,6,7,8,9,10} define the equivalence relation R on A
as follows : For all x,y A, xRy <=> 3|(x-y) . Find the
distinct equivalence classes of R(discrete math)

Let A={1,2,3,4,5,6} and let R be the relation on A defined by
xRy iff x + y is even and x is less than or equal to y. Determine
if R is partial order.

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

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?

Suppose we define the relation R on the set of all people by the
rule "a R b if and only if a is Facebook friends with b." Is this
relation reflexive? Is is symmetric? Is
it transitive? Is it an equivalence relation?
Briefly but clearly justify your answers.

Problem 3
For two relations R1 and
R2 on a set A, we define the
composition of R2 after R1
as
R2°R1 = { (x,
z) ∈ A×A | (∃ y)( (x,
y) ∈ R1 ∧ (y, z) ∈
R2 )}
Recall that the inverse of a relation R, denoted
R -1, on a set A is defined as:
R -1 = { (x, y) ∈
A×A | (y, x) ∈ R)}
Suppose R = { (1, 1), (1, 2),...

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