Modern Algebra
Set Theory (I)
Equivalence Relation

Let S be the set of all integers and let n > 1 be a fixed integer.
Define for a,b in S, a ~ b if a - b is a multiple of n.
Prove that this defines an equivalence relation on S.

The fully formatted problem is in the attached file.

Concerning discrete math, I am very confused as to the relationship between an equivalencerelation and an equivalence class.
I would very much appreciate it if someone could explain this relationship and give examples of each such that the relationship (or difference) is clear.

Let P, P' be equivalencerelations on a set A. Let n, n' be the number of equivalence classes of p, p', respectively.
A) define an equivalencerelation p'' as follows:
xp''y <=> (xpy) and (xp'y)
what is the least number of equivalence classes of p''? What is the greatest number of equivalence classes of p''?
B)defin

1. List the ordered pairs in the equivalencerelations produced by these partitions of {0,1,2,3,4,5}
a) {0}, {1,2}, {3,4,5}
b) {0,1}, {2,3}, {4,5}
c) {0,1,2}, {3,4,5}
d) {0}, {1}, {2}, {3}, {4}, {5}
2. Which of these collections of subsets are partitions of the set of integers?
a) the set of even integers and the set of

Show that the following are equivalent:
(a) ~ is an equivalencerelation on a group G
(b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a.
See the attached file.

Let X={1,2,3,4,5}, Y={1,2}.
Define relation R on g(x) by ARB iff AY =BY
*Note: g(x) is the power set of x and R is a equivalencerelation (no need to prove this)*
a) C={2,3}. List the elements of [C], the equivalence class containing C.
b) How many distinct equivalence classes are there?
c) Suppose X={1,2,...,n

For m, n, in N define m~n if m^2 ? n^2 is a multiple of 3.
(a.) Show that ~ is an equivalencerelation on N.
(b.) List four elements in the equivalence class [0].
c) List four elements in the equivalence class [1].
(d.) Are there any more equivalence classes. Explain your answer.

Regarding the set of natural numbers N, answer the following questions.
Is â?? a binary relation? Explain.
Is â?? reflexive? Explain.
Is â?? symmetric? Explain.
Is â?? transitive? Explain.
Is â?? an equivalencerelation? Explain.

Let A and B be finite subgroups of G.
Even though AB need not be a subgroup of G, show that
|AB||A n B| = |A||B|.
(Hint: Define (a_1,b_1)~(a_2,b_2) iff a_1b_1 = a_2b_2.
Prove that ~ is an equivalencerelation and examine the equivalence
classes.)
I need a detailed and rigorous proof to study for a test p