How do you express this relation between two sets?

Question

I have two sets: Set $A = \{20, 5\}$ and set $B = \{3, 8, 5, 4\}$. As you can see, the elements of set $A$ are sum and average of the elements of the set $B$. That means value of every element of $A$ depends on values of all of the elements of $B$. My question is, how do we express this dependency in a formal way? Thanks!

Answer 1

Iam not sure what you are looking for but $$a_1 = 20 = \sum_{j=1}^{|B|} b_j = 3+8+5+4$$ and $$a_2= 5 = \frac{1}{|B|} \sum_{j=1}^{|B|} b_j = \frac{a_1}{|B|} =5$$ where $|B|=4$ stands for the number of elements of the set $B$.

Answer 2

One way of viewing this dependency is considering two functions $s,a: \mathcal P_{fin} \left({\mathbb{R}}\right) \to \mathbb{R}$ such that given a finite subset $X \subset \mathbb{R}$ we have $s(X) = \sum_{x \in X}x$ and $a(X) = \frac{s(X)}{\lvert X \rvert}$. In your example we have $A = \{s(B)\} \cup\{a(B)\}$.

Answer 3

The problem is that "depends on" is an extremely vague notion.

• Does $4$ depend on $2$?

• Does $17$ depend on $2$?

• Does every element of $\{11, 12, 13\}$ depend on all of the elements of $\{2, 3\}$?

Let's look at that last question in particular. I wrote those two sets down first without thinking about them - so in that sense, the answer should be no. But after the fact, I can find a dependency! Namely, we have:

• $11=2^1+3^2$: given a set $\{a_1

• $12=3(3+2)-3(3-2)$: take the largest element, multiply it by the sum of the elements; and from that, subtract the largest element, multiplied by the diameter of the set (the difference between the smallest element and the largest element).

• And $13=2\cdot 2+3\cdot 3$, the sum of the squares of the elements.

Note that each of these three operations makes sense for any finite set - so $\{11, 12, 13\}$ does depend on $\{2, 3\}$!

The discrepancy is that the notion of "depends on" isn't just a relation between two sets, but rather between two definitions of sets: it is intensional, rather than extensional. We can try to formulate extensional versions, but they will differ from your intuition in various ways. For example, one version would be to say "$A$ depends on $B$ if every element of $A$ is definable from the elements of $B$" in the sense of model theory. That's great, except for two things:

• Definability only makes sense in a background structure - so we need to fix some structure (say, $\mathbb{N}$ with $+$ and $\times$) containing $A$ and $B$. That's easy to do, but different choices of structure will yield different notions of definability, so this isn't really fully extensional.

• Definability is more powerful than you might think! For instance, in $\mathbb{N}$ with $+$ and $\times$, every natural number is definable - so any set of natural numbers depends on any other set.