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10:06
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Q: Inductive definition of a set

GomathiI'm a beginner in set theory and I have doubt regarding mathematical induction. I came across the following examples. Example 1: Find the set given by the following definition: 1) $ 3 \in P $ 2) For $x,y \in P, x + y \in P $ 3) Only those elements obtained from steps (1) and (2) are in $P$ ...

elementary-set-theory induction
Daniel Pietrobon
Daniel Pietrobon
Which part do you not understand?
Gomathi
Gomathi
@DanielPietrobon How did they found that the set has multiples of 3 in example 1 and how to write the definitions in Example 2, Sir?
Daniel Pietrobon
Daniel Pietrobon
Well, what is 3+3?
Gomathi
Gomathi
@DanielPietrobon But, they didn't mention that y is also 3, Sir.
Daniel Pietrobon
Daniel Pietrobon
x and y are elements in P. 3 is an element in P by (1).
Gomathi
Gomathi
10:06
@DanielPietrobon So, initially as they have mentioned 3 belongs to P, we take both x and y as 3. Am I right, Sir?
Daniel Pietrobon
Daniel Pietrobon
Yes, exactly. :)
So now you know that 3 and 3+3=6 are in P.
What other numbers does P contain?
Gomathi
Gomathi
@DanielPietrobon now it has 3 and 6 sir
Daniel Pietrobon
Daniel Pietrobon
and what else?
Remember, for any elements x and y in P, x+y is also in P.
Gomathi
Gomathi
so, 3, 6, 9 ,12, etc
Daniel Pietrobon
Daniel Pietrobon
Very good! Is it clear that these are exactly the multiples of three?
Gomathi
Gomathi
10:16
yes sir. Thank you.
Daniel Pietrobon
Daniel Pietrobon
Great, now go back and see if Question 2 is clearer to you. :)
Good day Gomathi.
Oh, and by the way, I am no sir. ;)
Gomathi
Gomathi
@DanielPietrobon So, first time we take 3, then 3+3=6, then 3+6=9, then 9+6=15, etc. Am I correct?
Daniel Pietrobon
Daniel Pietrobon
Yes, but is enough to notice that 3 + 3 + 3 + ... + 3 (k times) is 3*k, a multiple of 3.
Gomathi
Gomathi
In the second example, it is given as both 2 and 3 because multiples of 2 or multiples of 3 alone doesn't make the set. So, initially they have given both. Am I correct?
Daniel Pietrobon
Daniel Pietrobon
Yes. That is a nice observation.
If we only had 2 we could only make 2,4,6,8,10,... etc.
These are all the even numbers.
How are we going to make all the odd numbers?
Gomathi
Gomathi
10:23
by giving multiples of 3
Daniel Pietrobon
Daniel Pietrobon
Does that work? What are the multiples of 3? Write the first few down.
Gomathi
Gomathi
3,6,9,...
Daniel Pietrobon
Daniel Pietrobon
So you are missing 5 and 7...
Gomathi
Gomathi
once 3 comes in there, we can form 5 using 3+2,...so we can get multiples of 5 also...am I correct?
Daniel Pietrobon
Daniel Pietrobon
Yes. Very good.
Gomathi
Gomathi
10:25
after 5, 5+2 for 7
Daniel Pietrobon
Daniel Pietrobon
By gosh, I think you've got it!
Gomathi
Gomathi
Yes.Also, why is it always x+y,...? Can't it be x*y or any other relations?
Daniel Pietrobon
Daniel Pietrobon
Because those are the rules of the game.
Gomathi
Gomathi
so, is it x+y always?
Daniel Pietrobon
Daniel Pietrobon
You might like to think of what would happen if you replaced they condition
x + y is in P
with x*y is in P
Gomathi
Gomathi
10:27
we will be missing certain elements
Daniel Pietrobon
Daniel Pietrobon
Indeed.
If we knew that 3 was in P.
and x*y was in P.
What elements would we get?
Gomathi
Gomathi
3,9,27,...
Daniel Pietrobon
Daniel Pietrobon
Very good. I think you understand the idea now. :)
Gomathi
Gomathi
we miss 6...
yes...thank you so much
to make it clear.
the first step is the initial set of elements
only based on the second step, we add the remaining elements to the initial step.
am i right?
Daniel Pietrobon
Daniel Pietrobon
Yes sir. ;)
Gomathi
Gomathi
10:31
Am a girl. :)
Can you kindly gather these together and post it as answer, so that I will accept it.
Gomathi
Gomathi
10:48
@DanielPietrobon ok brother. Thank you so much for your help.
good night

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