Solution Verification: An equivalence relation on finite subsets of $\mathbb{N}$, defined by having equal sums of the elements in the sets

Question

Let $E$ be the set whose elements are the finite parts (subset) of $\mathbb{N}$.

We define the binary relation $S$ on $E$ by posing $\forall A \in E, B \in E ,ASB $ if and only if $\sum\limits_{a\in A}a =\sum\limits_{b\in B}b $

Questions:

1. Give two examples of set couples $A$ and $B$ such as $ASB$.

2. Show that the binary relation $S$ is an equivalence relation.

My resolution:

1. Suppose that: $A=(1+3+2)$ and $B=(2+2+2)$ that satisfies the relation S which implies $ASB$.

2.So as we all know, according to the definition of equivalent relation the elements of the set $E$ must satisfy the next properties such as reflexivity, symmetricity and transitivity:

Reflexivity: Let $\forall a \in A$ and $A \in\mathbb{N} $ then

$a_1+a_2+a_3... = a_1+a_2+a_3...$

Hence $ASA$

Symmetricity: $\forall a \in A$,$\forall b \in B$ and $A,B \in\mathbb{N}$ also we suppose that $ASB$ which means $a_1+a_2+a_3... = b_1+b_2+b_3...$ then

$ASB \Longrightarrow a_1+a_2+a_3... = b_1+b_2+b_3...$

we also know that "=" possesses the commutative property then we have

$b_1+b_2+b_3... = a_1+a_2+a_3... \Longrightarrow BSA $

Transitivity:$\forall a \in A$,$\forall b \in B,\forall c \in C $ and $A,B,C \in\mathbb{N}$.Also we suppose that $ASB$ and $BSC$ which means $a_1+a_2+a_3... = b_1+b_2+b_3...$ and $b_1+b_2+b_3... = c_1+c_2+c_3...$ then we obtain:

\begin{equation} \left\{ \begin{aligned} a_1+a_2+a_3... = b_1+b_2+b_3\ldots\\ b_1+b_2+b_3... = c_1+c_2+c_3\ldots \end{aligned} \right. \end{equation}
$\Longrightarrow (a_1+a_2+a_3...)+(b_1+b_2+b_3...)=(b_1+b_2+b_3...)+(c_1+c_2+c_3...)$

$\Longrightarrow (a_1+a_2+a_3...)=(c_1+c_2+c_3...)$

$\Longrightarrow ASC$

I have written my own resolution to this problem, knowing that it contains some mistakes. For this reason I would like to get a correction. Thank you in advance!

Answer 1

Your solution to $(1)$ neglects something: that $B$ is just the set $\{2,2,2\}=\{2\}$; this is why you should be more careful and write things explicitly as sets. (We delete duplicate entries in sets.)

Of course, this is not difficult to rectify: just find another finite set of distinct positive integers which sum up to $6$, to be compatible with $A=\{1,2,3\}$. For instance, $\{6\}$ itself.

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Your solution to $(2)$ is fine in its spirit, I would just rewrite things to use less ambiguous notation and strange wording. For instance,

• Reflexivity: $\sum_{a \in A} a = \sum_{a \in A} a$, so $ASA$
• Symmetry: Observe that $$ASB \implies \sum_{a \in A} a = \sum_{b \in B} b \implies \sum_{b \in B} b = \sum_{a \in A} a \implies BSA$$ as desired.
• Transitivity: Note that, assuming $ASB$ and $BSC$, $$\sum_{a \in A} a = \sum_{b \in B} b \text{ and } \sum_{b \in B} b = \sum_{c \in C} c$$ and, since $=$ is a transitive relation, we have $\sum_{a \in A} a = \sum_{c \in C} c$, and so $ASC$.

Your wording can also use some improvements. The quantifiers you're using don't even necessarily reflect the definitions, the notations you're using, or what you want to look at. It is okay to use words.

For instance, I'd start off the transitivity argument by simply saying "Let $ASB$ and $BSC$ hold." The need for them to be arbitrary is already handled by this, and if you really want to be specific, you could say "Let $A,B,C \in E$ be such that $ASB$ and $BSC$ hold." This way you're not cluttering things up with quantifiers, and still getting the right level of generality.

Some other nitpicks:

• Your arguments for each suggest $A,B,C \in \mathbb{N}$. They are subsets, not elements; if anything, they're in $E$ since they're also meant to be finite.
• The reason I went for summation notation is because you know the summations are meant to be finite, whereas the trailing $\cdots$ can be read as implying infinite summation.
• You reference, for instance, $a \in A$ in your quantifiers but later are using $a_i$ to refer to elements of $A$. You should be consistent between the two (an issue rectified by using the summation notation).