Show that the relation $R$, consisting of all pairs $\left( x,y\right) $ where $x$ and $y$ are bit strings of length three or more that agree except perhaps in their first three bits, is an equivalence relation on the set of all bit strings.

Solution:

To show this, we need to show that the relation $R$ is reflexive, symmetric, and transitive.

Let $x,y,z\in R$ be bit strings that differ at most in their first three bits.

Reflexive: It is reflexive because after the first 3 bits, $x=x$

Symmetric: It is symmetric because after the first 3 bits, $x=y$ and $y=x$

Transitive: It is transitive because $x=y$ and $y=z$, therefore $x=z$

Since $R$ is reflexive, symmetric, and transitive, $R$ is an equivalence relation.

This document created by Scientific Notebook 4.0.