Question

To determine whether "the set \(\{ x + n\mid n \in Z\} \) for all \(x \in {(0,1)^{\prime \prime }}\) are the partitions of the set of real numbers.

Short answer

The set is the partition of the set of real numbers.

Step-by-step solution

Given data

The set \(\{ x + n\mid n \in Z\} \) for all \(x \in (0,1)\).

Concept used of partition of sets

The set \(S\) in a partition must be nonempty, pair wise disjoint, and have as their union

Find partition of set

The set \(\{ x + n\mid n \in Z\} \) for all \(x \in (0,1)\).

Here, the set consist all real numbers with a fixed fractional part.

Hence, the set is the partition of the set of real numbers.