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Two or more things are 'distinct' if no two of them are the same thing. In mathematics, two things are called 'distinct' if they are not equal.

Contents

Example

Proving distinctness

See also

Example

A quadratic equation over the complex numbers always has two roots.
The equation
: ''x''² − 3''x'' + 2 = 0
factors as
: (''x'' − 1)(''x'' − 2) = 0
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:
:''x''² − 2''x'' + 1 = 0
factors as
: (''x'' − 1)(''x'' − 1) = 0
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 is introduced to explain this.)

Proving distinctness

In order to prove 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 ''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 says (among other things) that distinct elements of a Banach space can be proved to be distinct using only linear functionals.

★ In category theory, if ''f'' is a functor between categories 'C' and 'D', then ''f'' always maps isomorphic objects to isomorphic objects. Thus, one way to show two objects of 'C' are distinct (up to isomorphism) is to show that their images under ''f'' are distinct (up to isomorphism).

See also

★ distinction

★ list of distinctions