Question

Show that the relation \({\rm{R}}\) consisting of all pairs \((x,y)\) such that \(x\) and \(y\) are bit strings of length three or more that agree in their first three bits is an equivalence relation on the set of all bit strings of length three or more.

Short answer

\(R\)is an equivalence relation.

Step-by-step solution

Given data

\(R = \{ (x,y)\mid x\)and\(y\) agree in their first three bits \(\} \)

Concept used of equivalence relation

An equivalence relation is a binary relation that is reflexive, symmetric and transitive.

Prove equivalence relation

Define a relation \(R\) on the set \(A\) (set of all bit strings of length three or more) as:

\(R = \{ (x,y)\mid x\)and\(y\) agree in their first three bits \(\} \)

Then, clearly \({\rm{R}}\) is reflexive, as for all \(x \in A\),

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

Now suppose \((x,y) \in R\), then it means that \({\rm{x}}\) and \({\rm{y}}\) agree in their first three bits.

But it is equivalent to that \(y\) and \(x\)agree in their first three bits.

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

So, \({\rm{R}}\) is symmetric.

Now, let \((x,y) \in R\) and \((y,z) \in R\)

This implies that \(x\)and yagree in their first three bits and that \(y\) and \(z\) agree in their first three bits.

This implies that \(x\) and \(z\) agree in their first three bits.

That is \((x,z) \in R\)

So, \(R\) is transitive.

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