1. ## Induction proof

Hello Everyone :-)

1)

I have to prove with Induction that if Αi⊆U for every positive integer I , where A is an omnium (or group don't remember the right English phrase =/ ) and U is the Whole, i have to prove that :

2)
We have a sequence of integers (c) that is for every
n>= 1 . Prove with induction that for every n>=0

3)

Again -.- , i have to prove with induction : for every n > = 1 .

Well , its not that i am trying to find easy answers on the web without trying at all on my own, but seriously i 've burned my mind already trying to solve them, and nothing happened...

2. Take 2), for example. First I would recommend writing a table
Code:
 0 |  1  |  2  |  3  |  4  |
---------------------------
c0 | c1  | c2  | c3  | c4  |
with several initial values of $\displaystyle c_i$. Does it fit the general formula? Can you see why, or is it a total surprise? I mean that in order to prove something one has to become comfortable with the concepts of that particular problem and have some intuition why the claim should be true.

If you did this, you already finished the base case. Now let $\displaystyle P(n)$ be the statement that $\displaystyle c_n=3^{2^n}$. The induction hypothesis is $\displaystyle P(n)$. What you need to prove is $\displaystyle P(n+1)$ -- be sure to write $\displaystyle P(n)$ and $\displaystyle P(n+1)$ explicitly. If at this point you don't see how to prove $\displaystyle P(n+1)$ from $\displaystyle P(n)$ and the definition of $\displaystyle c_{n+1}$, post here all of these things and describe your difficulty.

3. ## DeMorgan's laws

Originally Posted by primeimplicant
Hello Everyone :-)

1)

I have to prove with Induction that if Αi⊆U for every positive integer I , where A is an omnium (or group don't remember the right English phrase =/ ) and U is the Whole, i have to prove that :

2)
We have a sequence of integers (c) that is for every
n>= 1 . Prove with induction that for every n>=0

3)

Again -.- , i have to prove with induction : for every n > = 1 .

Well , its not that i am trying to find easy answers on the web without trying at all on my own, but seriously i 've burned my mind already trying to solve them, and nothing happened...

For the basis step the complement of $\displaystyle A_1$ = the complement of $\displaystyle A_1$ so the basis step is true.
The complement of [$\displaystyle A_1$ $\displaystyle \cup$ $\displaystyle A_2$ $\displaystyle \cup$ ... $\displaystyle \cup$ $\displaystyle A_k$ ] is equal to the intersection of the complements by DeMorgan's law.