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.


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.

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