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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:
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, 5/2 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 quantificationLogic In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything or every relevant thing. The resulting statement is a universally quantified statement, and we have univer. For example, the solution to example 2 above can be specified as:
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 equationIn 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. Many properties of functions can be determined by studying the types of functional equation. For example of this, consider
which says that f(x) * f(x) = f(x) for every value of x. If the universe of discourse consists of all functionsIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtu from the real lineIn mathematics, the real line is simply the set 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. The real line has been studied at least since the days of t 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 functionIn mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output, the function is said to bes from R to itself, then the solutions for f are only the constant functions with value one or zero.
See also: atomic sentence, compound sentence.
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