Question

Define the relation τ on Z by a τ b if and only if there exists x ∈ {1, 4, 16} such that ax ≡ b (mod 63).

(a) Prove that τ is an equivalence relation.

(b) Prove that there exists an integer n with 1 ≤ n ≤ 62 such that the equivalence class of n is {m ∈ Z | m ≡ n (mod 63)}.

Answer #1

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Please comment if needed.

Define the relation τ on Z by aτ b if and only if there exists x
∈ {1,4,16} such that
ax ≡ b (mod 63).
(a) Prove that τ is an equivalence relation.
(b) Prove that there exists an integer n with 1 ≤ n ≤ 62 such
that the equivalence class of n is{m ∈ Z | m ≡ n (mod 63)}.

Define a relation R on Z by aRb if and only if |a| = |b|.
a) Prove R is an equivalence relation
b) Compute [0] and [n] for n in Z with n different than 0.

Let R be the relation on Z defined by:
For any a, b ∈ Z , aRb if and only if 4 | (a + 3b). (a) Prove that
R is an equivalence relation.
(b) Prove that for all integers a and b, aRb if and only if a ≡
b (mod 4)

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

Consider the relation on the real numbers R. a ~ b if (a−b) ∈ Z.
(Z is the whole integers.)
1) Give two real numbers that are in the same equivalence
class.
2) Give two real numbers that are not in the same equivalence
class.
3) Prove that this relation is an equivalence relation.

Define a relation on N x N by (a, b)R(c, d) iff ad=bc
a. Show that R is an equivalence relation.
b. Find the equivalence class E(1, 2)

Define the relation S on RxR by (x,y)S(a,b) if and only if x^2 +
y^2= a^2 + b^2.
a) Prove S in an equivalence relation
b) compute [(0,0)], [(1,2)], and [(-3,4)].
c) Draw a picture in R^2 representing these three equivalence
classes.

1. Suppose we have the following relation defined on Z. We say
that a ∼ b iff 2 divides a + b. (a) Prove that the relation ∼
defines an equivalence relation on Z. (b) Describe the equivalence
classes under ∼ .
2. Suppose we have the following relation defined on Z. We say
that a ' b iff 3 divides a + b. It is simple to show that that the
relation ' is symmetric, so we will leave...

Let R be the relation of congruence mod4 on Z: aRb if a-b= 4k,
for some k E Z.
(b) What integers are in the equivalence class of 31?
(c) How many distinct equivalence classes are there? What are
they?
Repeat the above for congruence mod 5.

13. Let R be a relation on Z × Z be defined as (a, b) R (c, d)
if and only if a + d = b + c.
a. Prove that R is an equivalence relation on Z × Z.
b. Determine [(2, 3)].

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