# Equivalence Relations and Classes

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.

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#### Solution Preview

Please see the attachment.

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.