1

So, I'm unraveling the Halmos's book "Naive Set Theory", just for fun. But I stumbled upon the following problem:

"Let $\{I_j\}$, $j \in J$ be a family. Write $K = \displaystyle\bigcup_{j \in J} I_j$ and let $\{A_k\}$, $k \in K$, be a family. It's not difficult to prove that $$\displaystyle\bigcup_{k \in K} A_k = \displaystyle\bigcup_{j \in J}(\bigcup_{i \in I_j} A_i)$$ That's the generalization of the associative law for unions. (Until here I'm ok)

Formulate and prove the generalization of the commutative law"

I have no clue about how to start the formulation, could someone help me? Thanks in advance.

Jean Marie
  • 81,803
Fractal Admirer
  • 454
  • I have taken the liberty to modify your previous title "I need help..." that could be a title for all questions here. – Jean Marie Aug 31 '19 at 18:45

2 Answers2

1

Commutativity is that the order in which we take the union does not matter, and one way to formulate this is to look at permutations of the index set:

If $f: I \to I$ is a bijection then $\bigcup \{A_i: i \in I\} = \bigcup \{A_{f(i)}: i \in I \}$

E.g. taking $I=\{0,1\}$ and $f$ the swap function ($f(0)=1,f(1)=0$), $A_0 \cup A_1 = A_1 \cup A_0$ is a special case.

Henno Brandsma
  • 242,131
1

If $I$ and $J$ are non-empty sets and, for each pair $(i,j) \in I \times J$, there is a set $A_{ij}$ then $\bigcup_{i \in I}\bigcup_{j \in J}A_{ij}=\bigcup_{j \in J} \bigcup_{i \in I}A_{ij}$. This, I believe, is the commutativity of unions.

Kavi Rama Murthy
  • 311,013
  • That makes sense thank you – Fractal Admirer Sep 01 '19 at 17:33
  • Unfortunately this is just a particular case of the generalized associative law: consider the family ${I_j}{j \in J}$ such that it sends $j$ to the set of all ordered pairs of $I \times J$ whose second coordinate is $j$ then, by the generalized associative law, you can write: $$ \cup{(i,j) \in I \times J} A_{(i,j,)} = \cup_{j \in J} \cup_{(i,j) \in I_j} A_{(i,j)}$$ and the same can then be done starting with the family ${J_i}_{i \in I}$ defined analogously. – Matteo Menghini Jan 15 '23 at 14:55