Question

Which of the following are partitions of , the set of real numbers? Explain your answers.
a. {In : n ∈ ℤ}, where In = {x ∈ ℝ : n ≤ x ≤ n + 1}
b. {Jn : n ∈ ℤ }, where Jn = {x ∈ ℝ: n ≤ x < n + 1}

Answer 1

By defenition, a partition of a set $\mathbb R$ is a set of non-empty subsets of $\mathbb R$  such that every element $x$  in $\mathbb R$ is in exactly one of these subsets.

a. Let $\{I_n : n \in\mathbb Z\}$, where $I_n = \{x \in\mathbb R : n ≤ x ≤ n + 1\}$. It is not a partition of a set $\mathbb R$. Indeed, $I_0=[0,1]$ is a unique set that contains the real number $0.5$ and $I_1=[1,2]$ is a unique set that contains the real number $1.5$. On the other hand, a real number 1 belong to both set $I_0$ and $I_1.$

b. Let $\{J_n : n \in\mathbb Z\}$, where $J_n = \{x \in\mathbb R : n ≤ x < n + 1\}$. Then it is a partition of a set $\mathbb R$. Let $x$ be a real number.The floor function $\lfloor x \rfloor$ is defined to be the greatest integer less than or equal to the real number $x$. Let $n=\lfloor x \rfloor$. Then $x\in[n,n+1)$. Since $[n,n+1)$ and $[m,m+1)$ are disjoint for different integers $n$ and $m$, $\{J_n : n \in\mathbb Z\}$ is a partition.