Definition:Open Statement

Definition

Let $P$ be a statement.

$P$ is an open statement if and only if $P$ contains at least one free occurrence of a variables that appears in it.

Also known as

An open statement can also be seen referred to as an open sentence.

The word formula can also be seen.

Examples

Prime Number

The statement:

$x$ is a prime number

is an open statement, as the variable $x$ appears as a free occurrence.

Also see

• Definition:Closed Statement

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): open sentence
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): variable: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): open sentence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): variable: 2.