Open sentence
In
mathematics, an
open sentence is a
sentence in which there are specific
numbers which, when used to replace the variables, will allow the resulting
expression to evaluate to
true.
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 pairss of numbers that are additive inversess;
- 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, 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 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: compound sentence