Question

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.

Answer #1

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?

Let F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

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?

2. Let R be a relation on the set of integers ℤ defined by ? =
{(?, ?): a2 + ?2 ?? ? ??????? ??????} Is this
relation reflexive? Symmetric? transitive?

a)
Let R be an equivalence relation defined on some set A. Prove
using induction that R^n is also an equivalence relation. Note: In
order to prove transitivity, you may use the fact that R is
transitive if and only if R^n⊆R for ever positive integer n
b)
Prove or disprove that a partial order cannot have a cycle.

Determine the distance equivalence classes for the relation R is
defined on ℤ by a R b if |a - 2| = |b - 2|.
I had to prove it was an equivalence relation as well, but that
part was not hard. Just want to know if the logic and presentation
is sound for the last part:
8.48) A relation R is defined on ℤ by a R b if |a - 2| = |b -
2|. Prove that R...

Consider the relation R= {(1,2),(2,2),(2,3),(3,1),(3,3)}. Is R
transitive, not reflexive, symmetric or equivalence relation?

Construct a binary relation R on a nonempty set A satisfying the
given condition, justify your solution.
(a) R is an equivalence relation.
(b) R is transitive, but not symmetric.
(c) R is neither symmetric nor reflexive nor transitive.
(d) (5 points) R is antisymmetric and symmetric.

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.

A relation R is defined on Z by aRb if |a−b| ≤ 2. Which of the
properties reflexive, symmetric and transitive does the relation R
possess? Explain why If R does not possess one of these
properties,

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 8 minutes ago

asked 29 minutes ago

asked 44 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago