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Inductive

Specialty Definition: Inductive

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Specialty Definition: Inductive reasoning

(From Wikipedia, the free Encyclopedia)

Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which a general rule is inferred from some set of specific observations. It is to ascribe properties or relations to types based on limited observations of particular tokens; or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is used, for example, in using specific propositions such as:

• This swan is white.
• A billiard ball moves when struck with a cue.

to infer general propositions such as:

• All swans are white.
• For every action, there is an equal and opposite re-action

Some philosophers consider the term "inductive logic" a misnomer because the validity of inductive reasoning is not dependent on the rules of formal logic which is by definition only deductive, not inductive. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same validity as the initial assumptions. In the example above, the conclusion that all swans are white is obviously wrong, but may have been thought correct in Europe until the settlement of Australia. Inductive arguments are never binding but they may be cogent. Inductive reasoning expresses the truth-value of its inferences in terms of probability rather than necessity.

The problem of induction, the search for a justification for inductive reasoning, was formally addressed first by David Hume. Hume criticised induction based on repeated experiences.

Philosophers since at least David Hume recognized a significant distinction between two kinds of statements, later called by Immanuel Kant "analytic" and "synthetic."

• Analytic truths, such as "All bachelors are unmarried men," or "Human beings are two-legged animals" are supposed to be true by virtue of the meanings of the words alone.
• Synthetic statements, such as "All ravens are black," or "All men are mortal," are true if at all only by virtue of some facts about the world. One has to discover that men die and ravens are black.

W. V. Quine debunked this distinction in his influential essay Two Dogmas of Empiricism and postulated that any empirical evidence that seems to falsify any particular theory can always be accommodated by the theory in question. (See ontological relativity.)

Both statistics and the scientific method rely on both induction and deduction.

See also

• Deductive reasoning
• Explanation
• Logic
• Falsifiability

External link

• Inductive Reasoning

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Inductive reasoning."

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Inductor

(From Wikipedia, the free Encyclopedia)

An inductor is a passive electrical component that produces a voltage proportional to the instantaneous change in current flowing through it:

V = L × dI/dt,

where V is the voltage generated, dI/dt is the rate of change of current, and L is a property of the device called inductance. The SI unit of inductance is the henry (H).

Thus an inductor resists changes in current. A pure inductor does not offer any resistance to direct current (an actual one does slightly), except when the current is switched on and off, then it makes the change more gradual.

When a sinusoidal alternating current flows through an inductor, a sinusoidal alternating voltage (or electromotive force, abbr. emf) is induced. The amplitude of the emf is related to the amplitude of the current and to the frequency of the sinusoid by the following equation.

V = I × ωL

where ω is the angular frequency of the sinusoid defined in terms of the frequency f as

ω = 2πf

The term ωL is known as inductive reactance, which is denoted by the symbol X_L and is the positive imaginary component of impedance.

Construction

An inductor is usually constructed as a coil of conducting material, usually copper wire. A core of ferrous material is sometimes used.

This effect can be understood as follows: the current produces a magnetic field; a change in current gives a change of this magnetic field; a changing magnetic field causes an electromotive force in the conductor. An induction coil is closely related to electromagnets in structure, but used for a different purpose—to store energy in a magnetic field.

Smaller inductors used for very high frequencies are sometimes made with a wire passing through a ferrite cylinder or bead.

History

In 1885, William Stanley, Jr built the first practical induction coil based on Lucien Gaulard and Josiah Willard Gibbs' idea. It was the precursor of the modern transformer.

See also

Electricity, Electronics, Capacitor, Transformer

Synonyms

coil, induction coil, choke, reactor

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Inductor."

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Mathematical induction

(From Wikipedia, the free Encyclopedia)

Mathematical induction, or proof by induction, is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. It can also be used in more general settings as will be described below. An induction variant is used in computer science to prove that expressions which can be evaluated are equivalent, and this is known as structural induction.

The simplest and most common form of mathematical induction proves that a statement holds for all natural numbers n and consists of two steps:

1. Showing that the statement holds when n = 0.
2. Showing that if the statement holds for n = m, then the same statement also holds for n = m + 1.

To understand why the two steps are in fact sufficient, it is helpful to think of the domino effect: if you have a long row of dominos standing on end and you can be sure that

1. The first domino will fall.
2. Whenever a domino falls, its next neighbor will also fall.

then you can conclude that all dominos will fall.

Example

Suppose we wish to prove the statement:

for all natural numbers n. This is a simple formula for the sum of the natural numbers up to the number n. The proof that the statement is true for all natural numbers n proceeds as follows.

Proof

Check it is true for n = 0. Clearly, the sum of the first 0 natural numbers is 0, and 0.(0 + 1) / 2 = 0. So the statement is true for n = 0. We could define the statement as P(n), and thus we have that P(0) holds.

Now we have to show that if the statement holds when n = m, then it also holds when n = m + 1. This can be done as follows.

Assume the statement is true for n = m, i.e.,

Adding m + 1 to both sides gives

By algebraic manipulation we have

Thus we have

This is that statement for n = m + 1. Note that it has not been proved as true: we made the assumption that P( m ) is true, and from that assumption we derived P( m + 1 ). Symbolically, we have shown that:

However, by induction, we may now conclude that the statement P(n) holds for all natural numbers n:

1. We have P( 0 ), and thus P( 1 ) follows
2. With P( 1 ), P( 2 ) follows
3. ... etc

Generalizations

This type of proof can be generalized in several ways. For instance, if we want to prove a statement not for all natural numbers but only for all numbers greater than or equal to a certain number b then the following steps are sufficient:

1. Showing that the statement holds when n = b.
2. Showing that if the statement holds for n = m then the same statement also holds for n = m + 1.

This can be used, for example, to show that n² > 2n for n ≥ 3. Note that this form of mathematical induction is actually a special case of the previous form because if the statement that we intend to prove is P(n) then proving it with these two rules is equivalent with proving P(n + b) for all natural numbers n with the first two steps.

Another generalization allows that in the second step we not only assume that the statement holds for n = m but also for all n smaller than or equal to m. This leads to the following two steps:

1. Showing that the statement holds when n = 0.
2. Showing that if the statement holds for all n ≤ m then the same statement also holds for n = m + 1.

This can be used, for example, to show that fib(n) = [Φⁿ - (-1/Φ)ⁿ ] / 5^1/2 where fib(n) is the n^th Fibonacci number and Φ = (1 + 5^1/2) / 2 (the socalled Golden mean). Since fib(m + 1) = fib(m) + fib(m - 1) it is straightforward to prove that the statement holds for m + 1 if we can assume that it already holds for both m and m - 1. Also for this generalization it holds that it is in fact just a special case of the first form; let P(n) be the statement that we intend to prove then proving it with these rules is equivalent with proving the statement ' P(m) for all m ≤ n ' for all natural numbers n with the first two steps.

The last two steps can be reformulated as one step:

1. Showing that if the statement holds for all n < m then the same statement also holds for n = m.

This is in fact the most general form of mathematical induction and it can be shown that it is not only valid for statements about natural numbers, but for statements about elements of any well-founded set, that is, a set with a partial order that contains no infinite descending chains (where < is defined such that a < b iff a ≤ b and a ≠ b).

This form of induction, when applied to ordinals (which form a well-ordered and hence well-founded class), is called transfinite induction. It is an important proof technique in set theory, topology and other fields.

Proofs by transfinite induction typically need to distinguish three cases:

1. m is a minimal element, i.e. there is no element smaller than m
2. m has a direct predecessor, i.e. the set of elements which are smaller than m has a largest element
3. m has no direct predecessor, i.e. m is a so-called limit-ordinal

See also three forms of mathematical induction.

Proof or reformulation of mathematical induction

The principle of mathematical induction is usually stated as an axiom of natural numbers, see Peano axioms. However, it can be proved in some logical systems; for instance, if the following axiom:

The set of natural numbers is well-ordered

is employed.

Note that the additional axiom is indeed an alternative formulation of principle of mathematical induction. That is, the two are equivalent. See proof of mathematical induction.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Mathematical induction."

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