Conclusion for Induction (1 Viewer)

[Frie]

Is it necessary to write a conclusion after we prove the third step in induction.

What exactly do we write?

 

[Trebla]

It is necessary for completeness but it is not marked in the HSC (different story for schools marking though). In the HSC it is sufficient to write 'hence by induction...'. Personally I prefer to write something like 'Since the statement is true for n = 1, then by induction it is true for all positive integers n'

 

[Frie]

Thanks, now I don't have to worry about it

 

[QZP]

For a strict school, is this conclusion perfect (without flaws/ambiguities/etc)?
"Statement true for n = k+1 IFF true for n = k. Since statement true for n = n0 (base case), therefore true for all n >= n0 by mathematical induction"

Feel free to attack me

 

[obliviousninja]

I do. Since true for n=1, also must be true for n=2,3..., therefore true for all>=0

 

[SpiralFlex]

I don't see the need for more than because you've essentially done & stated it in the base step and inductive step.

"Hence by mathematical induction the proposition is true for all integers n > 0." Even though it is claimed to be not mark you should still do it in non-exam situations because it is good practice.

 

[Web Addict]

As my conclusion, I just write "Hence, true by PMI for all integers, n."

 

[QZP]

I am asking because my school REQUIRES the longer approach to the conclusion (and they are notorious for deducting marks for the smallest ambiguities or such).

 

[SpiralFlex]

I am asking because my school REQUIRES the longer approach to the conclusion (and they are notorious for deducting marks for the smallest ambiguities or such).

What do they want you to do?

 

[RealiseNothing]

For a strict school, is this conclusion perfect (without flaws/ambiguities/etc)?
"Statement true for n = k+1 IFF true for n = k. Since statement true for n = n0 (base case), therefore true for all n >= n0 by mathematical induction"

Feel free to attack me

Get rid of the IFF, doesn't work.

 

[RealiseNothing]

I do. Since true for n=1, also must be true for n=2,3..., therefore true for all>=0

oh really

 

[QZP]

Get rid of the IFF, doesn't work.

What do you mean it does not work? :S Just if then?

Edit: I did a quick google and it says something about "A iff B" implies that A is true only if B is true, and B is true if A is true. Or something like that. But the problem is that it's one sided and such we say "A if B" for the case of A: n = k+1 and B: n = k?

 

[RealiseNothing]

What do you mean it does not work? :S Just if then?

Edit: I did a quick google and it says something about "A iff B" implies that A is true only if B is true, and B is true if A is true. Or something like that. But the problem is that it's one sided and such we say "A if B" for the case of A: n = k+1 and B: n = k?

Just say if.

"n=k+1 is true IFF n=k is true" implies that for n=k+1 to be true then n=k MUST be true. However take the base case n=0. It is true, but n=-1 is not true, which is a contradiction.

 

[asianese]

I've seen some teachers and tutors try to be fancy pantsy and put IFF in lots of cases where the converse isn't true at all (some geometry proofs). Then students think its all cool and whack and IFF all over the place.. but they don't even know what it means.

 

[QZP]

Just say if.

"n=k+1 is true IFF n=k is true" implies that for n=k+1 to be true then n=k MUST be true. However take the base case n=0. It is true, but n=-1 is not true, which is a contradiction.

Oh damn. Thanks for clarifying

@asianese

I agree, but really I am just trying to progress my mathematical elegance. It might not be my strongest subject but I like best myself.

 

[SpiralFlex]

I don't believe iff is used in any paper HSC except in some books. You'll deal with it plentifully in later maths especially discrete

For now your proofs are either direct or via contradiction

 

[asianese]

Just don't use it.

 

[RealiseNothing]

I agree, but really I am just trying to progress my mathematical elegance. It might not be my strongest subject but I like best myself.

Mathematical elegance shouldn't come from throwing around cool terminology such as "IFF", nor should you purposely try to look elegant otherwise that elegance becomes forced and fake.

Elegance should come naturally with experience as your insight grows, and should come from the maths behind the proof itself. Think of it this way. Elegance of a proof should only be noticeable by some one who knows the maths used in the proof. If a non-maths random says it looks elegant, that's not really "elegance" but rather illusion.

Just my $0.02

 

[QZP]

Okay, thanks for valuable inputs

 

[mahmoudali]

watch out motherfuckers not always n>= 0
: D