Partition of a set A where $A= \{0,1\}^3$

Question

Let $A = \{0, 1\}^3$: that is, $A$ is the set of all ordered triples with entries from 0 and 1. Then define a relation $R \subseteq A \times A$ such that $xRy$ if and only if $x$ and $y$ have the same number of 0s. Note that $R$ is an equivalence relation. Give the partition of $A$ created by the equivalence classes of $R$.

I'm going through set partition theory and although I understand the definition of partition, I'm not sure how to be sure that all the partitions I have listed are truly all the partitions of a set.

And then I'm not quite sure how to begin thinking about this question. Can anyone help out?

Answer 1

The equivalence classes are $$\{(0,0,0)\}$$ $$\{(1,0,0),(0,1,0),(0,0,1)\}$$ $$\{(1,1,0), (1,0,1),(0,1,1)\}$$ $$\{(1,1,1)\}$$

Answer 2

Hint:

The equivalence classes naturally correspond to the possible numbers of $0$s, namely $0,1,2$ or $3$. So there is a partition in at most four subsets. (I say "at most" in case some of the subsets would be empty.)

Regroup $$000: 0,001: 1, 010: 1, 011: 2, 100: 1,101: 2, 110: 2, 111: 3.$$