Open sentence

In mathematics, an open sentence is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined (and hence the sentences are no longer regarded as "open"). These possible replacement values are assumed to range over a subset of either the real or complex numbers, depending on the equation or inequality under consideration (in applications, real numbers are usually associated also with measurement units). The replacement values which produce a true equation or inequality are called solutions of the equation or inequality, and are said to "satisfy" it.

In mathematical logic, a non-closed formula is a formula which contains free variables. (Note that in logic, a "sentence" is a formula without free variables, and a formula is "open" if it contains no quantifiers, which disagrees with the terminology of this article.) Unlike closed formulas, which contain constants, non-closed formulas do not express propositions; they are neither true nor false. Hence, the formula

$x$ is a number

(1)

has no truth-value. A formula is said to be satisfied by any object(s) such that if it is written in place of the free variable(s), it will form a sentence expressing a true proposition. Hence, "5" satisfies (1). Any sentence which results from a formula in such a way is said to be a substitution instance of that formula. Hence, "5 is a number" is a substitution instance of (1).

Mathematicians have not adopted that nomenclature, but refer instead to equations, inequalities with free variables, etc.

Such replacements are known as solutions to the sentence. An identity is an open sentence for which every number is a solution.

Examples of open sentences include:

1. 3x − 9 = 21, whose only solution for x is 10;
2. 4x + 3 > 9, whose solutions for x are all numbers greater than 3/2;
3. x + y = 0, whose solutions for x and y are all pairs of numbers that are additive inverses;
4. 3x + 9 = 3(x + 3), whose solutions for x are all numbers.
5. 3x + 9 = 3(x + 4), which has no solution.

Example 4 is an identity. Examples 1, 3, and 4 are equations, while example 2 is an inequality. Example 5 is a contradiction.

Every open sentence must have (usually implicitly) a universe of discourse describing which numbers are under consideration as solutions. For instance, one might consider all real numbers or only integers. For example, in example 2 above, 1.6 is a solution if the universe of discourse is all real numbers, but not if the universe of discourse is only integers. In that case, only the integers greater than 3/2 are solutions: 2, 3, 4, and so on. On the other hand, if the universe of discourse consists of all complex numbers, then example 2 doesn't even make sense (although the other examples do). An identity is only required to hold for the numbers in its universe of discourse.

This same universe of discourse can be used to describe the solutions to the open sentence in symbolic logic using universal quantification. For example, the solution to example 2 above can be specified as:

For all x, 4x + 3 > 9 if and only if x > 3/2.

Here, the phrase "for all" implicitly requires a universe of discourse to specify which mathematical objects are "all" the possibilities for x.

The idea can even be generalised to situations where the variables don't refer to numbers at all, as in a functional equation. For an example of this, consider

f * f = f,

which says that f(x) * f(x) = f(x) for every value of x. If the universe of discourse consists of all functions from the real line R to itself, then the solutions for f are all functions whose only values are one and zero. But if the universe of discourse consists of all continuous functions from R to itself, then the solutions for f are only the constant functions with value one or zero.

References

• Definition at The Math Resource
• The Math Forum @ Drexel