Let L be a subset of {a,b}*
Define a relation R (R sub L) on S* as follows:
L

for All of x, y is a member of S*,
(x,y) are members of R if for all of z, xz are members of L iff yz are members of L

A) Show that R is an equivalence relation

B) Suppose L={a^i b^i where i >= 0}
What can you say about the index of R (number of classes)? is it finite or infinite?
Show some classes and elements in these classes to justify your answer?

c) Suppose L={a^i b^j where i,j >= 0}
What can you say about the index of R (number of classes)? is it finite or infinite?
Show some classes and elements in these classes to justify your answer?

D) Suppose L={a^i b^3i where i >= 0}
What can you say about the index of R (number of classes)? is it finite or infinite?
Show some classes and elements in these classes to justify your answer?

Note: ^ means to the power, so a^i means a to the power of i.

First, let's clarify the definitions.
is a set of sequences of 's and 's. If we can define multiplications for and , then . is a subset of . The relation defined ...

Solution Summary

Equivalence relations and classes are investigated. The solution is well presented.

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

Verify that each of the following are equivalencerelations on the plane R^2 (where R are real numbers) and describe the equivalenceclasses geometrically.
1) (x1,y1)R(x2,y2) if and only if x1 = x2
2) (x1,y1)R(x2,y2) if and only if x1 + y1 = x2+y2
3) (x1,y1)R(x2,y2) if and only if
x1^2 + y1^2 = x2^2 + y2^2.

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 equivalence relation (no need to prove this)*
a) C={2,3}. List the elements of [C], the equivalence class containing C.
b) How many distinct equivalenceclasses 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 equivalence relation 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 equivalenceclasses. Explain your answer.

I have two small problems. I need all the work shown and in the second problem please answer in detail and NOT just yes or no.
(See attached file for full problem description)

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 equivalence relation and examine the equivalenceclasses.)
I need a detailed and rigorous proof to study for a test p

See attached
Let R be the relation on the set {1,2} defined by 2R2 and the relation R holds for no other ordered pair except the pair (2,2). Show that R has exactly two of the three defining properties of an equivalence relation.

Let (X,B,mu) be a complete, finite measuable space. For each C,D in B, set
d(C,D) = mu (C / D)
where C / D is the symmetric difference of C and D. We say that two measurable sets C,D are equivalent if d(C,D)=0 (this is an equivalence relation).
Let E be the set of equivalenceclasses, and show that d introduces a metri