Question

What are the equivalence classes for the relation congruence modulo 6? Please explain in detail.

Answer #1

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.

Prove: Proposition 11.13. Congruence modulo n is an equivalence
relation on Z :
(1) For every a ∈ Z, a = a mod n.
(2) If a = b mod n then b = a mod n.
(3) If a = b mod n and b = c mod n, then a = c mod n

Please solve in full detail! Use the fact that
Zp*, the nonzero residue classes modulo a
prime p, is a group under multiplication to establish Wilson’s
Theorem.

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...

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?

1.
a. Consider the definition of relation. If A is the set of even
numbers and ≡ is the subset of ordered pairs (a,b) where a<b in
the usual sense, is ≡ a relation? Explain.
b. Consider the definition of partition on the
bottom of page 18. Theorem 2 says that the equivalence classes of
an equivalence relation form a partition of the set. Consider the
set ℕ with the equivalence relation ≡ defined by the rule: a≡b in ℕ...

What is the advantage of financing with debt compared to equity?
Please explain in detail.

What is a model organism? How is Arabidopsis a good model?
Please explain in detail.

What is a Finite Population Correction Factor?
Please explain its concept in detail

Let's say we have the following relation defined on the set {0,
1, 2, 3}:
{ (0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3) }
- Please answer the following 3 questions about this relation. (The
relation will be repeated for each question.) Is this relation a
function? Why or why not?
- What are the three properties that must be present in an
equivalence relation? Please give the names of the three properties...

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