jhonyy9
  • jhonyy9
- let a,b >=1 and n>=3 numbers from the set of natural numbers N , - how can prove it that always for any value of n exist numbers a and b such that this equation a+b+1=n is true ? - is this prove with reductio ad absurdum correct ? in a reducto ad absurdum argument, we assume the opposite is true. in this case, that would be: for some natural number n >=3 there are no natural numbers a >=1, b >=1 with a+b+1 = n. we don't know what n may be (except that it is greater or equal 3). since n >=3, n - 1 >=2. since n - 1 >=2, (n - 1)/2 >=1. (n - 1)/2 + (n - 1)/2 + 1 = (n - 1) + 1 = n.
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
Unfortunately, (n-1)/2 is not a natural number for every n. If it were, you would have demonstrated that every number greater than or equal to three is odd. You'll do much better using induction rather than reduction.
jhonyy9
  • jhonyy9
in a reducto ad absurdum argument, we assume the opposite is true. in this case, that would be: for some natural number n >=3 there are no natural numbers a >=1, b >=1 with a+b+1 = n. we don't know what n may be (except that it is greater or equal 3). since n >=3, n - 1 >= 2. since n - 1 >=2, (n - 1)/2 >=1. (n - 1)/2 + (n - 1)/2 + 1 = (n - 1) + 1 = n. but if n is a natural number, so is n - 1, and so is (n - 1)/2. therefore we have proved that for this n >=3, the natural numbers (n - 1)/2 don't exist. this is absurd. (the logical conclusion therefore being that no such natural number >=3 actually exists, so our assumption was wrong, so EVERY natural number >=3 has such an a and b >=1 with a+b+1 = n).
anonymous
  • anonymous
Let n be 4. Then n-1 is 3. (n-1)/2 is 3/2 which is not a natural number.

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anonymous
  • anonymous
There are two reasons to want to do this by induction. The first is that it's trivially easy. The second is that induction is how most theorems about natural numbers are proved.
jhonyy9
  • jhonyy9
- can you writing please one prove with induction ?
anonymous
  • anonymous
Normally induction would start with n=1 but the theorem concerns n>=3 so that's our first point. Is the theorem true for n=3? Can you write a+b+1 =3 for some values of a and b?
anonymous
  • anonymous
Don't overthink this. I said it was trivial.
jhonyy9
  • jhonyy9
1+1+1=3
anonymous
  • anonymous
Exactly. Now we assume that the theorem is true for every value up to n, and we test whether it's true for n+1. So assuming we can write n = a+b+1, can we write n+1 as the sum of two numbers plus 1?
jhonyy9
  • jhonyy9
n+1=a+b+1+1
anonymous
  • anonymous
Exactly. And the right hand side can be grouped so that it is two numbers plus 1 rather than three plus one as you've written it. Right?
jhonyy9
  • jhonyy9
like n+1=a+(b+1)+1 - ?
anonymous
  • anonymous
Again correct. You can bludgeon this one by examining cases, but it's much more useful and aesthetic as an exercise in induction.
jhonyy9
  • jhonyy9
can you help me please how ?
anonymous
  • anonymous
What further help would you like?
jhonyy9
  • jhonyy9
one complete math induction prove what will may be acceptebilly
anonymous
  • anonymous
You've done it, but let me repeat. To prove that the theorem is true by induction, we prove that it's true for the lowest value (3) and then prove that if it's true for n, it is also true for n+1.
anonymous
  • anonymous
For n=3, we see that n=1+1+1 so it works for a=1 and b=1.
anonymous
  • anonymous
Now assume that it's true for all values up to. This means that n=a+b+1 for some numbers a and b. Then n+1 = a+b+1+1 = a+(b+1)+1 and we have our two numbers, a and b+1 satisfying the equation.
anonymous
  • anonymous
Since the set of natural numbers is generated by starting with 1 and adding 1 repeatedly, this process covers every possible natural number. QED by induction.
jhonyy9
  • jhonyy9
- so and thats all ?
anonymous
  • anonymous
I said it was trivial.
anonymous
  • anonymous
You can also do it by the two possible cases. Any number n greater or equal to three is either even or odd. If even, there is some number a, such that n=a+a. Since n>3, a>1 so a-1>=1. so N=a+(a-1)+1.
anonymous
  • anonymous
If n is odd, then n-1 is even, so n-1 = a+a, so n=a+a+1 and the two values are a and a.
jhonyy9
  • jhonyy9
ok thank you very much - but i will come back again tomorrow
anonymous
  • anonymous
I'll watch for you.

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