Question

What are the equivalence classes of the bit strings in Exercise 30 for the equivalence relation \({R_4}\) from Example 5 on the set of all bit strings? (Recall that bit strings s and t are equivalent under \({R_4}\) if and only if they are equal or they are both at least four bits long and agree in their first four bits.)

Short answer

The equivalence class for the bit string is found to be an equivalence relation for \(\left\{ {{R_4}} \right\}\).

Step-by-step solution

 Given data.

The equivalence classes for bit string are as follows,

\(\left( a \right){\rm{ }}010,{\rm{ }}\left( b \right){\rm{ }}1011,{\rm{ }}\left( c \right){\rm{ }}11111,{\rm{ }}\left( d \right){\rm{ }}01010101\)

 Condition for equivalence relation.

A relation is defined on a set\(S\)is known as an equivalence relation if set is reflexive, symmetric and transitive.

If\(R\)is an equivalence relation on a set \(S\). Then, the set of all elements that are related to any element\(s\)of set\(S\)is known as the equivalence class of\(s\).

Define condition for the given relation for the set.

Let\(R\)is an equivalence relation defined on set\(S\)such that\(s \in S\).

\((s) = \{ x \in S\mid (s,x) \in R\} \)is known as an equivalence class of\(s\).

Let\(S\)is the set of all bit strings.

Thus, the relation\({R_4}\)is an equivalence relation on\(S\).

Where condition for the relation may be defined as:

\({R_5} = \left\{ {\begin{array}{*{20}{l}}{(x,y)\mid x{\rm{ and }}y{\rm{ are equal or they are both at least }}}\\{{\rm{ four bits long and agree in their first four bits }}}\end{array}} \right\}\)

 (a) Evaluation of equivalence class for the bit string (\({\rm{0}}{\bf{10}}\)).

Let\(s = 010 \in S\).

\(\begin{array}{c}(s) = \left\{ {x \in S:(s,x) \in {R_5}} \right\}\\ = \left\{ {\begin{array}{*{20}{l}}{x \in S\mid s = x{\rm{ or they are both at least four bits }}}\\{{\rm{ long and agree in their first four bits }}}\end{array}} \right\}\\ = \{ 010\} {\rm{ Here, the length of }}s{\rm{ is less than 4}}.\end{array}\)

Therefore, the equivalence class of the bit string is \(\{ 010\} \) in \(\left\{ {{R_4}} \right\}\).

 (b) Evaluation of equivalence class for the bit string (\(1011\)).

Let\(s = 1011\)in\(S\).

\(\begin{array}{c}(s) = \left\{ {x \in S:(s,x) \in {R_4}} \right\}\\ = \left\{ {\begin{array}{*{20}{c}}{x \in S\mid s = x{\rm{ or they are both at least four bits }}}\\{{\rm{ long and agree in their first four bits }}}\end{array}} \right\}\\ = \{ x\mid a = x\} {\rm{ Here, the length of }}s{\rm{ is less than }}4.\end{array}\)
Therefore, the equivalence class of the bit string is \(\{ 1011\} \) in \(\left\{ {{R_4}} \right\}\).

(c)  Evaluation of equivalence class for the bit string (\(11111\)).

Let\(s = 11111 \in S\)

\(\begin{array}{c}(s) = \left\{ {x \in S:(s,x) \in {R_4}} \right\}\\ = \left\{ {\begin{array}{*{20}{c}}{x \in S\mid s = x{\rm{ or they are both at least four bits }}}\\{{\rm{ long and agree in their first four bits }}}\end{array}} \right\}\\ = \left\{ {\begin{array}{*{20}{l}}{x \in S\mid x{\rm{ has a bit string of length at least four bits }}}\\{{\rm{ and the first four bits are }}1,1,1,1}\end{array}} \right\}\end{array}\)

Therefore, the equivalence class of the bit string is\(\{ 1111\} \)in\(\left\{ {{R_4}} \right\}\).

 (d) Evaluation of equivalence class for the bit string (\({\bf{01010101}}\)).

Let\(s = 01010101 \in S\)

\(\begin{array}{c}(s) = \left\{ {x \in S:(s,x) \in {R_4}} \right\}\\ = \left\{ {\begin{array}{*{20}{l}}{x \in S\mid s = x{\rm{ or they are both at least four bits }}}\\{{\rm{ long and agree in their first four bits }}}\end{array}} \right\}\\ = \left\{ {\begin{array}{*{20}{l}}{x \in S\mid x{\rm{ has a bit string of length at least four bits }}}\\{{\rm{ and the first fore bits are }}0,1,0,1}\end{array}} \right\}\end{array}\)

Therefore, the equivalence class of the bit string is \(\{ 0101\} \) in \(\left\{ {{R_4}} \right\}\).