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# STEP Tricks and Hacks Thread watch

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1. (Original post by Lucasium)
A STEP teacher at my college mentioned a 'loop' technique, he didn't go into any detail because he said it's not needed for A level. Does anybody know what it is?
What topic?

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2. (Original post by drandy76)
What topic?

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It was during FP3, it could have been to do with differentiation.
3. (Original post by Lucasium)
It was during FP3, it could have been to do with differentiation.
I can't say I've ever heard of a loop technique. There are loops in computing of course, as well as graph theory and an algebraic structure known as a loop, though I can't tell you much about them - I imagine wikipedia can.

A function that starts and ends at the same space is often called a loop, for obvious reasons, which is part of a more general topological definition.
The only thing I can think of that's even vaguely related to differentiation would be the directional derivative, but that's a stretch.
4. (Original post by joostan)
I can't say I've ever heard of a loop technique. There are loops in computing of course, as well as graph theory and an algebraic structure known as a loop, though I can't tell you much about them - I imagine wikipedia can.

A function that starts and ends at the same space is often called a loop, for obvious reasons, which is part of a more general topological definition.
The only thing I can think of that's even vaguely related to differentiation would be the directional derivative, but that's a stretch.
Thanks anyway, I'll ask him later and if it's useful I'll post in here
5. (Original post by Lucasium)
Thanks anyway, I'll ask him later and if it's useful I'll post in here
Oh I thought of something that's loop like, though not related to differentiation.
If asked to show a chain of equivalences, i.e that it suffices to show that .
Obviously this extends beyond statements in the logical way.
This is a neat trick often used in proofs of theorems with multiple parts to the statement, though possibly not directly relevant to this thread.
6. (Original post by Lucasium)
Thanks anyway, I'll ask him later and if it's useful I'll post in here
(Original post by joostan)
Oh I thought of something that's loop like
Loops with IBP, i.e: 2 IBP's and then solving to find I. It's not really a trick though... more like standard knowledge.
7. (Original post by Zacken)
Loops with IBP, i.e: 2 IBP's and then solving to find I. It's not really a trick though... more like standard knowledge.
A good point, I hadn't considered that.
8. (Original post by Zacken)
Loops with IBP, i.e: 2 IBP's and then solving to find I. It's not really a trick though... more like standard knowledge.
It's also in the A-Level syllabus (for Edexcel at least) so wrong on all counts there
9. (Original post by joostan)
Oh I thought of something that's loop like, though not related to differentiation.
If asked to show a chain of equivalences, i.e that it suffices to show that .
Obviously this extends beyond statements in the logical way.
This is a neat trick often used in proofs of theorems with multiple parts to the statement, though possibly not directly relevant to this thread.
If any STEP question required that it would be much too difficult I think. It's a very neat trick though, useful for algebra courses at uni!
10. (Original post by joostan)
Oh I thought of something that's loop like, though not related to differentiation.
If asked to show a chain of equivalences, i.e that it suffices to show that .
Obviously this extends beyond statements in the logical way.
This is a neat trick often used in proofs of theorems with multiple parts to the statement, though possibly not directly relevant to this thread.
Wait, this is boggling me - how does this work? You get logical equivalence between a number of statements just by proving implications from one to another?
11. (Original post by Zacken)
Wait, this is boggling me - how does this work? You get logical equivalence between a number of statements just by proving implications from one to another?
Yeah, basically.

To prove C implies B, note C implies A and A implies B hence you're done. Because you already have B implies C, you've proved B iff C. Similarly for the other statements, and hence you've proved they're all equivalent.
12. (Original post by joostan)
Oh I thought of something that's loop like, though not related to differentiation.
If asked to show a chain of equivalences, i.e that it suffices to show that .
Obviously this extends beyond statements in the logical way.
This is a neat trick often used in proofs of theorems with multiple parts to the statement, though possibly not directly relevant to this thread.
This is very strong. Damn

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13. (Original post by shamika)
Yeah, basically.

To prove C implies B, note C implies A and A implies B hence you're done. Because you already have B implies C, you've proved B iff C. Similarly for the other statements, and hence you've proved they're all equivalent.
Oh, wow - yes! Thank you.
14. (Original post by shamika)
Yeah, basically.

To prove C implies B, note C implies A and A implies B hence you're done. Because you already have B implies C, you've proved B iff C. Similarly for the other statements, and hence you've proved they're all equivalent.
(Original post by joostan)
Oh I thought of something that's loop like, though not related to differentiation.
If asked to show a chain of equivalences, i.e that it suffices to show that .
Obviously this extends beyond statements in the logical way.
This is a neat trick often used in proofs of theorems with multiple parts to the statement, though possibly not directly relevant to this thread.
That's pretty cool
15. (Original post by Student403)
That's pretty cool
Engineers don't find this cool.

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16. (Original post by physicsmaths)
Engineers don't find this cool.

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Do mathmos love physics almost as much as, if not more than maths? ;p
17. (Original post by Zacken)
Oh, wow - yes! Thank you.
So a standard Cambridge move is to use a long loop of these implications in your lecture notes to prove the equivalence of a number of statements - perhaps a dozen.

Then, on the example sheet, you are asked to prove the equivalence of number 2 and number 5 on the list. You soon realize that trailing through the sequence of implications is long and tedious and there must be a more direct proof. Your supervisor grins at you with an evil grin.

Then on the tripos paper, you have to prove the equivalence of numbers 2 and 8 on the list...
18. (Original post by Student403)
Do mathmos love physics almost as much as, if not more than maths? ;p
A true mathmo loves physics more than maths.

Proof:

Isaac Newton was a true mathmo, and he loved physics more than maths. QED
19. (Original post by atsruser)
A true mathmo loves physics more than maths.

Proof:

Isaac Newton was a true mathmo, and he loved physics more than maths. QED
physicsmaths
20. (Original post by Gregorius)
So a standard Cambridge move is to use a long loop of these implications in your lecture notes to prove the equivalence of a number of statements - perhaps a dozen.
Is this "long loop" terminology standard? I've never heard of it before.

The first time I saw it was in one of those god-awful linear algebra proofs about the equivalence of "Rank A = n" and "unique solutions of Ax=b" and about 1/2 a dozen other things. It struck me as too clever-clever at the time, and I'm not sure that I really believed the proof - it looked like stuff had been missed out.

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Updated: February 17, 2018
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