Question

"Prove that, for the equivalence classes [a,b] and operations + and x: the distributive law for x over + is true"

Answer #1

For each of the following, prove that the relation is an
equivalence relation. Then give the information about the
equivalence classes, as specified.
a) The relation ∼ on R defined by x ∼ y iff x = y or xy = 2.
Explicitly find the equivalence classes [2], [3], [−4/5 ], and
[0]
b) The relation ∼ on R+ × R+ defined by (x, y) ∼ (u, v) iff x2v
= u2y. Explicitly find the equivalence classes [(5, 2)] and...

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...

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.

Abstract Algebra I
Corollary 1.26- Two equivalence classes of an
equivalence relation are either disjoint or equal.
Corollary 2.11- Let a and b be two integers
that are relatively prime. Then there exist integers r and s such
that ar+bs=1.
PLEASE ANSWER THE FOLLOWING:
1) Why is Corollary 1.26 true?
2) Why is Corollary 2.11 true?

9e) fix n ∈ ℕ. Prove congruence modulo n is an equivalence
relation on ℤ. How many equivalence classes does it have?
9f) fix n ∈ ℕ. Prove that if a ≡ b mod n and c ≡ d mod n then a
+ c ≡b + d mod n.
9g) fix n ∈ ℕ.Prove that if a ≡ b mod n and c ≡ d mod n then ac
≡bd mod n.

On set R2 define ∼ by writing(a,b)∼(u,v)⇔ 2a−b =
2u−v. Prove that∼is an equivalence relation on R2
In the previous problem:
(1) Describe [(1,1)]∼. (That is formulate a statement P(x,y)
such that [(1,1)]∼ = {(x,y) ∈ R2 | P(x,y)}.)
(2) Describe [(a, b)]∼ for any given point (a, b).
(3) Plot sets [(1,1)]∼ and [(0,0)]∼ in R2.

Consider the following relation ∼ on the set of integers
a ∼ b ⇐⇒ b 2 − a 2 is divisible by 3
Prove that this is an equivalence relation. List all equivalence
classes.

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.

Let H be a subgroup of a group G. Let ∼H and ρH be the
equivalence relation in G introduced in class given by
x∼H y⇐⇒x−1y∈H, xρHy⇐⇒xy−1 ∈H.
The equivalence classes are the left and the right cosets of H in
G, respectively. Prove that the functionφ: G/∼H →G/ρH given
by
φ(xH) = Hx−1
is well-defined and bijective. This proves that the number of
left and right cosets are equal.

Given a preorder R on a set A, prove that there is an
equivalence relation S on A and a partial ordering ≤ on A/S such
that [a] S ≤ [b] S ⇐⇒ aRb.

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