Question

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.

Answer #1

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.

the relation R on the set of all people where aRb means that a
is younger than b. Determine if R is:
reflexive
symmetric
transitive
antisymmetric

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.

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.

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.

Suppose that ~ is a relation defined on the vertices of a graph
G. There are three things we have to check to show that u ~ v is an
equivalence relation: that it is reflexive, symmetric and
transitive. Describe clearly what each one requires.

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

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 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...

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...

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