Relation Question for Discrete Mathematics

Question

Let $X\subseteq\mathbb Z$ and define a relation $\tau$ on the power set of $X$ as follows:

$\forall A,B\in\wp(X)[A\tau B\iff\text{the sum of the elements in }A\text{ equals the sum of the elements in }B]$.

a)Prove that $\tau$ is an equivalence relation.

b)Let $X = \{0,1,2,3\}$. Write down the partition of $\wp(X)$ given by the equivalence classes of $\tau$ for this set $X$.

I know that for part a) to prove its a equivalence relation i need to prove that both A&B are reflexive, Symmetric and Transitive. But i am not to sure how do i prove it specifically. While for part b) of the question I totally don't understand at all. Thank you for teaching me!

Answer 1

I preassume that the sum of elements in $X$ is well defined and finite.

Define $s:\wp(X)\to\{0,1,2,\dots\}$ by stating that: $$s(A)=\sum_{a\in A}a$$

Then: $$A\tau B\iff s(A)=s(B)$$

So

• $\tau$ is reflexive $\iff\forall A\in\wp(X)[s(A)=s(A)]$ for all $A\in\wp(\mathcal Z)$
• $\tau$ is symmetric $\iff\forall A,B\in\wp(X)[s(A)=s(B)\implies s(B)=s(A)]$
• $\tau$ is transitive $\iff\forall A,B,C\in\wp(X)[s(A)=s(B)\wedge s(B)=s(C)\implies s(A)=s(C)]$

Observe that it is very obvious now that $\tau$ is an equivalence relation.

The partition is: $$\{\{A\in\wp(X)\mid s(A)=k\}\mid k\in\{0,1,2,\dots\}\}$$This allows you to answer (b).