Question

Let
X be finite set . Let R be the relation on P(X). A,B∈P(X) A R B Iff
|A|＝|B| prove R is an equivalence relation

Answer #1

Let S be a finite set and let P(S) denote the set of all subsets
of S. Define a relation on P(S) by declaring that two subsets A and
B are related if A and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

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: Prove that R is an equivalence
relation.
you may use the following lemma: If p is prime and p|mn, then
p|m or p|n

A relation R on a set A is called circular if for all a,b,c in
A, aRb and bRc imply cRa. Prove that a relation is an equivalence
relation iff it is reflexive and circular.

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...

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
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?

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 R be an equivalence relation defined on some set A.
Prove using mathematical induction that R^n is also an
equivalence relation.

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.

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.

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