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- Open_sentence abstract "In mathematics, an open sentence (usually an equation or equality) 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</dl>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 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: 3x − 9 = 21, whose only solution for x is 10; 4x + 3 > 9, whose solutions for x are all numbers greater than 3/2; x + y = 0, whose solutions for x and y are all pairs of numbers that are additive inverses; 3x + 9 = 3(x + 3), whose solutions for x are all numbers. 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 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.".
- Open_sentence wikiPageExternalLink 53280.html.
- Open_sentence wikiPageExternalLink open_sentence.html.
- Open_sentence wikiPageID "228361".
- Open_sentence wikiPageRevisionID "595238140".
- Open_sentence hasPhotoCollection Open_sentence.
- Open_sentence subject Category:Logical_expressions.
- Open_sentence subject Category:Mathematics_education.
- Open_sentence subject Category:Propositional_calculus.
- Open_sentence type Abstraction100002137.
- Open_sentence type Appearance104673965.
- Open_sentence type Attribute100024264.
- Open_sentence type Countenance104679549.
- Open_sentence type Expression104679738.
- Open_sentence type LogicalExpressions.
- Open_sentence type Quality104723816.
- Open_sentence comment "In mathematics, an open sentence (usually an equation or equality) 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").".
- Open_sentence label "Open sentence".
- Open_sentence label "Sentença aberta".
- Open_sentence label "جملة مفتوحة (رياضيات)".
- Open_sentence label "开放句子".
- Open_sentence sameAs Sentença_aberta.
- Open_sentence sameAs m.01h7p0.
- Open_sentence sameAs Q830486.
- Open_sentence sameAs Q830486.
- Open_sentence sameAs Open_sentence.
- Open_sentence wasDerivedFrom Open_sentence?oldid=595238140.
- Open_sentence isPrimaryTopicOf Open_sentence.