Question

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)

Answer #1

A relation is equivalence relation if it is transitive, reflexive and symmetric

A relation is called reflexive if xRx for all possible values of x.

A relation is called symmetric if xRy then yRx should be possible.

A relation is called transitive if xRy and yRz then x should be related to z.

Here option (b) contains 3 numbers 1,1.8 and 3

It doesn't contains relation like (1, 1) (3, 3)...

This the given relation is not reflexive which means it is not equivalence relation.

Option (d) contains number like 1 and 1.5

It contains (1, 1) but doesn't contain (1.5, 1.5). Thus the given relation is not reflexive which makes it non equivalence relation.

There fore the answer is option (b) and (d)

If you have any questions comment down and please? upvote thanks

Let R be a relation on set RxR of ordered pairs of real numbers
such that (a,b)R(c,d) if a+d=b+c. Prove that R is an equivalence
relation and find equivalence class [(0,b)]R

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.

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?

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?

Let
A be the set of all real numbers, and let R be the relation "less
than." Determine whether or not the given relation R, on the set A,
is reflexive, symmetric, antisymmetric, or transitive.

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.

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,...

I have a discrete math question.
let R be a relation on the set of all real numbers
given by cry if and only if x-y = 2piK for some integer K. prove
that R is an equivalence relation.

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.

2. Define a relation R on pairs of real numbers as follows: (a,
b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a
partial order? Why or why not? If R is a partial order, draw a
diagram of some of its elements.
3. Define a relation R on integers as follows: mRn iff m + n is
even. Is R a partial order? Why or why not? If R is...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 31 minutes ago

asked 35 minutes ago

asked 38 minutes ago

asked 42 minutes ago

asked 42 minutes ago

asked 49 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago