Of the form
Encyclopedia
In mathematics
, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs
.
Statement:
The product of any two even
natural numbers is also even.
Proof:
Any even natural number is of the form 2n, where n is any natural number. Therefore, let us assume that we have two even numbers which we will denote by 2k and 2l. Their product is (2k)(2l) = 4(kl) = 2(2kl). Since 2kl is also a natural number, the product is even.
Note:
In this case, both exhaustivity
and exclusivity
were needed. That is, it was not only necessary that every even number is of the form 2n (exhaustivity), but also that every expression of the form 2n is an even number (exclusivity). This will not be the case in every proof, but normally, at least exhaustivity is implied by the phrase of the form.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
.
Example of use
Here is a proof which should be appreciable with limited mathematical background:Statement:
The product of any two even
Even and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
natural numbers is also even.
Proof:
Any even natural number is of the form 2n, where n is any natural number. Therefore, let us assume that we have two even numbers which we will denote by 2k and 2l. Their product is (2k)(2l) = 4(kl) = 2(2kl). Since 2kl is also a natural number, the product is even.
Note:
In this case, both exhaustivity
Proof by exhaustion
Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds...
and exclusivity
Exclusive
Exclusive typically means not with other things or not including other things.The terms inclusive and exclusive are contrasting terms, and often appear in the same context to describe contrasting things.Exclusive may refer to:...
were needed. That is, it was not only necessary that every even number is of the form 2n (exhaustivity), but also that every expression of the form 2n is an even number (exclusivity). This will not be the case in every proof, but normally, at least exhaustivity is implied by the phrase of the form.