Partition of Real Numbers

Question

Let R be the set of all real numbers. Is $\{\mathbb R^+,\mathbb R^−,\{0\}\}$ a partition of $\mathbb R$? Explain your answer.

My answer is no because of $\{0\}$. I am confused with $\{0\}$. please help.

Answer 1

The answer is yes because the union of 3 sets are $R$ and 3 sets are disjoint from each other. ${0}$ is just one point set of $0$.

Answer 2

It is a partition. In order that $\{A, B, C\}$ be a partition of some set $S$, it is necessary and sufficient that

• $A\cup B\cup C=S$, and
• $A\cap B = B\cap C = A\cap C=\varnothing$, and
• $A$, $B$, and $C$ are non-empty.

And similarly for other numbers of sets than three. (The second condition is that no two of the sets intersect; generally there would be more pairs of sets than there are sets when there are more than three sets.)

So yes, it is a partition.