Why does induction work

And what are we going to assume? Well, let us assume that this property holds for the immediate next component of the structure, therefore "unwrapping" one Succ. As said before, to assume can in some sense be viewed as "to actually call the thing", therefore we can create a small inline helper call like so:.

We cannot use Refl yet because clearly rhs is not lhs. In Idris we say:. Therefore we end up with:. Which is a constructive proof that the equality holds. Notice that we simply called the function in question with something, which is our way of assuming things. We can also assume we have x : Nat because the function was called with some x! The entire correspondence between propositions and types and further programs as proofs, proof normalization as program execution is known as Curry-Howard Isomorphism.

In explanations of the principle of induction, I think the 'domino' example is often not explored enough for it to be intuitive. Let me endeavour a more understandable explanation using the domino analogy.

As always, you first prove that P 1 is true or P 2 or P 3 or P 4 septillion , depending from where you want your hypothesis to be valid.

If you don't care that P 1 be true, you can begin from P 2 or wherever. Since P 3 is true, P 4 is true. Since P , is true, P , is true. The notion of the inductive hypothesis used to scare and confuse me until I was able to understand more clearly the underlying mechanism which makes use of it.

Basically, as I understand it, we are trying to perform a direct proof of an implication. One way to prove an implication is to assume the original statement is true and to see what must logically follow. It is my perception that this crucial preliminary explanation as to why we are making any kind of assumption or hypothesis is missing from the majority of explanations of how proof by induction works.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Dominoes and induction, or how does induction work? Ask Question. Asked 10 years, 9 months ago. Active 5 months ago. Viewed 12k times.

Call this the induction hypothesis. What happens when you push over the first domino? Add a comment. Active Oldest Votes. QED This is a perfectly valid proof! The proof is by strong induction, and it involves no algebraic manipulations whatsoever. Arturo Magidin Arturo Magidin k 49 49 gold badges silver badges bronze badges. RSS feed for comments on this post. TrackBack URI. You are commenting using your WordPress. You are commenting using your Google account.

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Create a free website or blog at WordPress. College Math Teaching March 6, The Principle of Mathematical Induction: why it works Filed under: induction , logic , transfinite induction , Uncategorized — collegemathteaching pm. What is mathematical induction? It is a common proof technique. Basically, if one wants to show that a statement is true in generality and that one can index the set of statements via the integers or by some other appropriate index set , then one can use induction. Here is a common example: suppose one wants to show that for all positive integers for example,.

An eddy current is a current that circulates in conductors, and has been induced in a conductor by a varying magnetic field. The current is created by changing the magnetic field. Eddy currents can heat up a metal object without touching them, they are also well known for stopping trains and rollercoasters.

They can be found in other objects such as electric brakes, car speedometer and energy meters, so you may be familiar with where eddy currents can be found. Electrical induction is also called magnetic induction.

Electromagnetic induction is the movement of magnets around a coil of wire that then create an electrical current through the wire. This can be done by rotating magnets while a coil of wire moves between them. The magnets will rotate between north and south poles, if the north pole of two magnets were to be pushed together they would repel and try to move away from each other.

But if south and north poles were to come together they would pull, as they are opposites and therefore attract. It is this push and pull that causes the magnets to rotate around the coil of wire, producing an electrical current. Due to the strong electromagnetic field created by an induction hob this type of hob could affect your pacemaker if you have one fitted. According to The British Heart Foundation BHF pacemaker wearers should keep a distance of at least 60cm or 2ft from the appliance as it may interfere with your settings.

The BHF have said that an induction hob would not cause significant damage to your device, but due to there being many different types of pacemakers, each could be affected in a different way. Skip to main content. Sequences and Series. Search for:. Mathematical Inductions. Learning Objectives Discuss what is meant by a sequence of mathematical statements.

Key Takeaways Key Points A sequence is an ordered list of objects or events. Like a set, it contains members, but unlike a set, the order of the members matters. A sequence of statements refers to the progression of logical implications of one statement.

Sequences of statements are important for mathematical inductions, which rely on infinite sequences of statements.