Giving an equivalence relation that corresponds to set partitions

Question

My question is:

Give equivalence relation that corresponds to the partitions A1 = {1,3,5} A2 = {2} A3 = {4,6}

of the set A = {1,2,3,4,5,6}

I don't know what the format of the relation should be, in fact I am confused about what an equivalence relation looks like.

Would it be R = {(1,1), (2,2), (3,3)........ so on}?

How do we determine a relation?

Answer 1

$$a,b\in A\;,\;\;\;aRb\iff a,b\in A_1\;\;or\;\;a,b\in A_2\;\;or\;\;a,b\in A_3\iff$$

$$R=\left\{(1,1),(2,2),...,(6,6), (1,3),(3,1),(1,5),(5,1), (3,5), (5,3) (4,6),(6,4)\right\}$$

Answer 2

The partitioning of $A$ into subsets/cells induces (and is induced by) the following relation $$R = \{(1, 1), (2, 2), (3,3), (4, 4), (5, 5), (6, 6), (1, 3), (3, 1), (1, 5), (5, 1), \\ (3, 5), (5, 3), (4, 6), (6, 4) \}$$

such that for any $a, b\in A$, $(a, b) \in R$ if and only if $a$ and $b$ are in the same subset/cell of the partition.

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N.B.

Typically, a partition on a set $A$ (note that "partition" is singular) refers to the entire family of what we call cells of the partition, such that the cells (in this case, subsets of $A:$ namely $A_1, A_2, A_3$) are pairwise disjoint and such that the union of the cells/subsets is exactly $A$.

I simply add the note above because it seems that you are using the term in a non-standard manner, incorrectly referring to $A_1, A_2, A_3$ as partitions, when they, together, constitute a partition on $A$.

Answer 3

You want the equivalence relation $R$ such that $(a,b) \in R$ iff $a, b \in A_i$ for some $i$.

In general, partition $\left\{A_i\right\}_{i \in I}$ corresponds to the equivalence relation $$R = \bigcup_{i\in I} A_i \times A_i$$i.e. the union of Cartesian squares of each set in the partition.