MathDB

(a)

Prove that

\mathbb{N}

can be partitioned into three (mutually disjoint) sets such that, if

m,n \in \mathbb{N}

and |m\minus{}n| is

2

5

, then

m

and

n

are in different sets.

(b)

Prove that

\mathbb{N}

can be partitioned into four sets such that, if

m,n \in \mathbb{N}

and |m\minus{}n| is

2,3,

5

, then

m

and

n

are in different sets. Show, however, that

\mathbb{N}

cannot be partitioned into three sets with this property.

Problem Statement