Question

Prove the following: Theorem. Let X be a set and {X_{i}
⊆ X : i ∈ I} be a partition of X. Then R = { (x_{1},
x_{2}) ∈ X × X : ∃i ∈ I,(x_{1} ∈ Xi) ∧
(x_{2} ∈ X_{i}) } is an equivalence relation on
X.

Answer #1

Let A be a non-empty set. Prove that if ∼ defines an equivalence
relation on the set A, then the set of equivalence classes of ∼
form a partition of A.

Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define
d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|.
Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…).
Prove that maps X onto the Cantor set and satisfies
(1/3)*d(x,y)≤|f(x)−f(y)|≤d(x,y) for x,y∈2N.

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

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

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.

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,...

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

Let x1, x2, x3 be real numbers. The mean, x of these three
numbers is defined
to be
x = (x1 + x2 + x3)/3
.
Prove that there exists xi with 1 ≤ i ≤ 3 such that
xi ≤ x.

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?

Prove that (Orbit-Stabilizer Theorem). Let G act on a finite set
X and fix an x ∈ X.
Then |Orb(x)| = [G : Gx] (the index of Gx).

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