Open sentence
Encyclopedia
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 proposition
s; they are neither true nor false. Hence, the formula
(1) x is a number
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:
Example 4 is an identity
.
Examples 1, 3, and 4 are equation
s, 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 number
s or only integer
s.
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 number
s, 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:
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
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 function
s from R to itself, then the solutions for f are only the constant function
s with value one or zero.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, a non-closed formula is a formula which contains free variables. (Note that in logic, a "sentence
Sentence (mathematical logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...
" 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
Constant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
, non-closed formulas do not express proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s; they are neither true nor false. Hence, the formula
(1) x is a number
has no truth-value. A formula is said to be satisfied by any object
Object (philosophy)
An object in philosophy is a technical term often used in contrast to the term subject. Consciousness is a state of cognition that includes the subject, which can never be doubted as only it can be the one who doubts, and some object or objects that may or may not have real existence without...
(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 pairOrdered pairIn mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
s of numbers that are additive inverseAdditive inverseIn mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
s; - 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
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
.
Examples 1, 3, and 4 are equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s, while example 2 is an inequality. Example 5 is a contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
.
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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s or only integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
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 number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, 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
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
using universal quantification
Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
.
For example, the solution to example 2 above can be specified as:
- For all x, 4x + 3 > 9 if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
.
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
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R to itself, then the solutions for f are all functions whose only values are one and zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
.
But if the universe of discourse consists of all continuous function
Continuous function
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". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s from R to itself, then the solutions for f are only the constant function
Constant function
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...
s with value one or zero.