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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: Image File history File links Question_book-3. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In modern philosophy, logic and linguistics, a proposition is what is asserted as the result of uttering a declarative 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). In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... In philosophy, an object is a thing, an entity, or a being. ...

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. In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In mathematics, the term 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. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... This article is about inequalities in mathematics. ...

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. The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The integers are commonly denoted by the above symbol. ... 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. ...

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: Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... 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. ...

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. ↔ ⇔ ≡ logical symbols representing iff. ...

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 In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...

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. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the real line is simply the set of real numbers. ... This article is about the number one. ... For other senses of this word, see zero or 0. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

See also

• atomic sentence
• compound sentence

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. In propositional calculus and in predicate calculus, an atomic sentence is an atomic formula which contains no variables. ... A compound sentence can refer, in similar ways, to two things In mathematical logic, sentences formed using logical operators to connect two. ...