Question

What is the equivalence class of the bit string 011 for the equivalence relation in Exercise 25?

Short answer

Equivalence class of the bit string \(011\) is the set of all bit string which contains exactly two \(1s\).

Step-by-step solution

  Given Data.

The bit string is given as \(011\) and s and t contains the same number of \(1s\).

Condition for equivalence relation.

A relation\(R\)on a set A is

(i) Reflexive, if\((a,a) \in R\)for all\((a \in A)\).

(ii) Symmetric, if\((a,b) \in R\)implies that\((b,a) \in R\).

(iii) Transitive, if\((a,b) \in R\)and\((b,c) \in R\)implies that\((a,c) \in R\).

(iv) An equivalence relation if it is reflexive, symmetric and transitive.

And for an equivalence relation\(R\)on the set\(A\), the equivalence class of the element\(a \in A\)is\({(a)_R} = \{ s\mid (a,s) \in R\} \).

Obtain the given relation as equivalence relation.

Let A be the set of all bit strings

Now, define a relation R on the set A as:

\(R = \{ (s,t)\mid s\)and \(t\)contain the same number of\(1s\)}.

Now for any\(x \in A\),It is self-evident that bit strings x and the same bit string x have the same number of 1s.

Thus,\((x,x) \in R\).

So, the relation\(R\)is reflexive.

Now, consider relation\((x,y) \in R\).

This demonstrates that bit string x and bit string y both contain the same number of 1s.

This is the same as saying that bit string y and bit string x both have the same number of 1s.

\((y,x) \in R\)

So, relation\(R\)is symmetric.

Now, consider a relation \((x,y) \in R\)and\((y,z) \in R\).

This means that bit strings x and y both have the same number of 1s, and bit strings y and z both have the same number of 1s.

This means that the bit strings x and z both have the same number of 1s.\((x,z) \in R\)

So, is\(R\)transitive.

Since, \(R\) is reflexive, symmetric and transitive. Therefore, \(R\) is an equivalence relation on the set \(A\) (set of all bit strings).

  Determine that the bit string \({\rm{0}}{\bf{1}}1\) contains only two \(1s\).

Consider the bit string\(a = 011\), whose equivalence class is to be found.

Then, by above definition of equivalence class,

\(\begin{aligned}{}{(a)_R} &= \{ x \in A\mid (a,x) \in R\} \\ &= \{ x \in A\mid a{\rm{ and }}x{\rm{ contain the same number of }}1\;{\rm{s}}\} \\ &= \{ x \in A\mid x{\rm{ contain two1s }}\} \end{aligned}\)

Hence, equivalence class of the bit string 011 is the set of all bit string which contains exactly two \({\rm{1s}}\).