Hello, I am having trouble proving the following theorem:
If A(1)....A(n) are finite pairwise disjoint sets then A(1) U A(2)U.....U A(n)= A(1)+....+A(n)
Are you translating into English?
That makes no sense in mathematics.
There is a theorem about counting. The symbol $\displaystyle \|A\|$ stands for the number of elements is the finite set $\displaystyle A$.
If $\displaystyle A_n:n=1,2,\cdots,K$ is a collection of pairwise disjoint finite sets then $\displaystyle \displaystyle \left\| {\bigcup\limits_n {A_n } } \right\| = \sum\limits_{n = 1}^K {\left\| {A_n } \right\|} $.
Is that what is meant by the notation?
That is where my problem lies in the induction proof:
I know to begin with the case n=2 so proving the basic counting principle.
Then I assume true for for n and take an n+1 element set and prove its true.
I am just struggling to write out a formal proof
Suppose that $\displaystyle \{A_n:n=1,2,\cdots,K\}$ is a collection of pairwise disjoint finite sets and it is thue that $\displaystyle \displaystyle \left\| {\bigcup\limits_n {A_n } } \right\| = \sum\limits_{n = 1}^K {\left\| {A_n } \right\|} $.
Now lets prove for $\displaystyle K+1$, so say $\displaystyle \{A_n:n=1,2,\cdots,K+1\}$ is a collection of pairwise disjoint finite sets [/tex].
Observe that $\displaystyle \bigcup\limits_{n = 1}^{K + 1} {A_n } $ is just $\displaystyle \left( {\bigcup\limits_{n = 1}^K {A_n } } \right) \cup A_{K + 1} $
Let $\displaystyle \mathbf{C}=\left( {\bigcup\limits_{n = 1}^K {A_n } } \right) $.
Note that $\displaystyle \mathbf{C} \cap A_{K+1}=\emptyset$.
Apply the basic Counting Principle.
Is that a statement? Or is it a question?
Why not learn to post in symbols? You can use LaTeX tags
[tex]\|A\|[/tex] gives $\displaystyle \|A\| $