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# Prove that the sum, difference and the product of two integers is also an integer. Watch

1. So I have an idea of how to do this, but it seems a bit two tedious and im sure there is a more elegant way. I'm looking at question 9 below:

My idea would be to look at different cases. Like we start by letting a and b be integers. Then we look at the different cases depending on the signs of a and b.

Problem is with the fact that they need to use the previous exercise, which is also in the picture above.

Any ideas?
2. I think continuing with induction would be the way to go. Results likes these seem so trivial, that it sometimes makes proving it seem harder than proving more difficult results.
3. (Original post by B_9710)
I think continuing with induction would be the way to go. Results likes these seem so trivial, that it sometimes makes proving it seem harder than proving more difficult results.
Yeh but you would have to do induction up and down for all integers.

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4. (Original post by gagafacea1)
So I have an idea of how to do this, but it seems a bit two tedious and im sure there is a more elegant way. I'm looking at question 9 below:

My idea would be to look at different cases. Like we start by letting a and b be integers. Then we look at the different cases depending on the signs of a and b.

Problem is with the fact that they need to use the previous exercise, which is also in the picture above.

Any ideas?
Prove that n+m n,mEN is a nat number.
Then without induction on integers.
Fact is that, if you have a difference of integers or sum it is either a natual number or
-(natural number) so proving the sum or difference is essentially the same thing wlog. That is if you can assum that if a is an integer then -a is an integer. If you canmt assume that, i can't be bothered to try anything else

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5. (Original post by B_9710)
I think continuing with induction would be the way to go. Results likes these seem so trivial, that it sometimes makes proving it seem harder than proving more difficult results.
(Original post by physicsmaths)
Prove that n+m n,mEN is a nat number.
Then without induction on integers.
Fact is that, if you have a difference of integers or sum it is either a natual number or
-(natural number) so proving the sum or difference is essentially the same thing wlog. That is if you can assum that if a is an integer then -a is an integer. If you canmt assume that, i can't be bothered to try anything else

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To be honest I still don't see how I could do that. I guess my biggest problem is that I don't know what I can assume to be defined and what I can't. Like how deep should I go with the proof? I've done all of the problems below except for the last:

This is how integers are defined in the book:
Attachment 558217558219
Attached Images

6. What module is this?
7. (Original post by Chittesh14)
What module is this?
It's not an A-Level module.
8. (Original post by Zacken)
It's not an A-Level module.
No... as in what module does this content come under?
9. (Original post by Chittesh14)
No... as in what module does this content come under?
Number Theory, it's a course - not a module.
10. (Original post by Zacken)
Number Theory, it's a course - not a module.
Oh lmao.....

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