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Mathematics-Online course: Prepcourse Mathematics - Basics - Propositional Logic

Statement

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A statement is a verbal expression that is either true or false, i.e. that is uniquely assigned one of the two truth values t (true) or f (false) respectively.

We use capital letters as symbols for statements, such as

$\displaystyle A:$   description$\displaystyle \quad . $

Statements can be joined using logical operators. Basic mathematical statements that cannot be derived from other statements are called axioms.
(Authors: Höllig/Kimmerle/Abele)
The following examples illustrate the notion of a mathematical statement.

The statement

$ A$:    Each natural number is a product of primes
is a true statement.

The statement

$ B$:    All primes are odd
is false, since $ 2$ is an even prime.

The hitherto unproved conjecture

$ C$:    There are an infinite number of twin primes
is a mathematical statement, since it is either true or false. Note that a twin prime is a pair of adjacent odd primes, such as

$\displaystyle (3,5),\, (5,7),\, (11,13),\ldots \,,$    for example$\displaystyle . $

The largest known twin prime so far (as on 19.4.2006) is $ 16869987339975 \cdot 2^{171960}\pm 1$.

However, the sentence

$ D$:    Friday the 13th is an unlucky day
is not a statement, since it cannot be assigned a truth value.

(Authors: Höllig/Hörner/Abele)
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  automatically generated 9/18/2007