iPedia Free Information Center

Information about Open Sentence

An open sentence is a sentence which contains variables. Unlike an ordinary sentence, which contains constants, open sentences do not express propositions; they are neither true nor false. Hence, the open sentence:

(1) x is a number

Has no truth-value. An open sentence is said to be satisfied by any object(s) such that if it is written in place of the variable(s), it will form a sentence expressing a true proposition. Hence, "5" satisfies (1). Any sentence which resembles an open sentence in form is said to be a substitution instance of that sentence. 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.
Example 4 is an identity. Examples 1, 3, and 4 are equations, while example 2 is an inequality.

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 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.

See also

An open sentence (usually an equation or inequality) 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" them.
variable (IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression.
..... Click the link for more information.
In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed.

Unspecified constants

The most widely mentioned sort of constant
..... Click the link for more information.
proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
..... Click the link for more information.
In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is true.

In classical logic, the only possible truth values are true and false.
..... Click the link for more information.
object is a thing, an entity, or a being. This may be taken in several senses.

In its weakest sense, the word object is the most all-purpose of nouns, and can replace a noun in any sentence at all.
..... Click the link for more information.
In mathematics, an ordered pair is a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or
..... Click the link for more information.
In mathematics, the additive inverse, or opposite, of a number n is the number that, when added to n, yields zero. The additive inverse of n is denoted −n.
..... Click the link for more information.
identity has several important uses:
  • An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions.

..... Click the link for more information.
equation is a mathematical statement, in symbols, that two things are the same (or equivalent). Equations are written with an equal sign, as in
.


The equation above is an example of an equality: a proposition which states that two constants are equal.
..... Click the link for more information.
inequality is a statement about the relative size or order of two objects. (See also: equality)
  • The notation means that a is less than b and
  • The notation means that a is greater than b.

..... Click the link for more information.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
..... Click the link for more information.
The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
..... Click the link for more information.
In mathematics, a complex number is a number of the form


where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
..... Click the link for more information.
Symbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical logic.
..... Click the link for more information.
In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing.
..... Click the link for more information.
“Iff” redirects here. For other uses, see IFF.
If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements
..... Click the link for more information.
In mathematics or its applications, a functional equation is an equation expressed in terms of both independent variables and unknown functions, which are to be solved for.
..... Click the link for more information.
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
..... Click the link for more information.
In mathematics, the real line is simply the set R of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space.
..... Click the link for more information.
This article is about the number one. For the year AD 1, see 1. For other uses, see 1 (disambiguation).

0 1 2 3 4 5 6 7 8 9 →

..... Click the link for more information.
0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
..... Click the link for more information.
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
..... Click the link for more information.
In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed.

Unspecified constants

The most widely mentioned sort of constant
..... Click the link for more information.
true or false under an interpretation which assigns values to the logical variables. We might for example make the following assignments:

Individual Constants
  • a: Socrates
  • b: Plato
  • c: Aristotle
Predicates:

..... Click the link for more information.
Compound sentence can mean:
  • In mathematical logic, a sentence consisting of two sentences joined by a logical operator.
  • In linguistics, a sentence consisting of multiple independent clauses joined by coordinating conjunctions and/or punctuation; see

..... Click the link for more information.


This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.
Herod_Archelaus


page counter