Distinct
Encyclopedia
Two or more things are distinct if no two of them are the same thing. In mathematics
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...

, two things are called distinct if they are not equal.

Example

A quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

 over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s sometimes has two roots.

The equation



factors
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

 as



and thus has as roots x = 1 and x = 2.
Since 1 and 2 are not equal, these roots are distinct.

In contrast, the equation:



factors as



and thus has as roots x = 1 and x = 1.
Since 1 and 1 are (of course) equal, the roots are not distinct; they coincide.

In other words, the first equation has distinct roots, while the second does not. (In the general theory, the discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

 is introduced to explain this.)

Proving distinctness

In order to prove
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...

 that two things x and y are distinct, it often helps to find some property that one has but not the other.
For a simple example, if for some reason we had any doubt that the roots 1 and 2 in the above example were distinct, then we might prove this by noting that 1 is an odd number while 2 is even.
This would prove that 1 and 2 are distinct.

Along the same lines, one can prove that x and y are distinct by finding some function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 f and proving that f(x) and f(y) are distinct.
This may seem like a simple idea, and it is, but many deep results in mathematics concern when you can prove distinctness by particular methods. For example,
  • The Hahn–Banach theorem
    Hahn–Banach theorem
    In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...

     says (among other things) that distinct elements of a Banach space
    Banach space
    In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

     can be proved to be distinct using only linear functional
    Linear functional
    In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

    s.
  • In category theory
    Category theory
    Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

    , if f is a functor
    Functor
    In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

     between categories
    Category (mathematics)
    In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

     C and D, then f always maps isomorphic
    Morphism
    In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

     objects to isomorphic objects. Thus, one way to show two objects of C are distinct (up to
    Up to
    In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

     isomorphism
    Isomorphism
    In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

    ) is to show that their images under f are distinct (i.e. not isomorphic).
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