Open Sentence
Pre-AlgebraAn open sentence is a mathematical statement containing one or more variables that is neither true nor false until specific values are substituted for the variables.
Definition
An open sentence is a math statement with a variable that is not yet true or false. Once you plug in a number, it becomes either true or false.
Example
$x + 3 = 10$ is an open sentence. Is it true? It depends on $x$. When $x = 7$, it is true. When $x = 4$, it is false.
Key Insight
It is called "open" because the question is still open, like an unanswered question waiting for the right value to close it.
Definition
An open sentence is an equation or inequality containing one or more variables. It has no definite truth value until the variables are replaced with specific values. The values that make it true form the solution set.
Example
$2x - 1 > 5$ is an open sentence (an open inequality). It becomes true for all $x > 3$ and false otherwise. The solution set is $\{x : x > 3\}$.
Key Insight
The contrast is with a closed sentence (a statement that is simply true or false): "$5 + 3 = 8$" is a closed sentence (true). An open sentence is always a template waiting to be filled.
Definition
In formal logic, an open sentence (or open formula) is a well-formed formula containing at least one free variable. It is not a proposition (which must have a fixed truth value) but becomes one upon binding all free variables with quantifiers or substituting values. Solving an equation is equivalent to finding a valuation that closes the formula to "true."
Example
The formula $P(x)$: $x^2 = 4$ is open. Closing it by universal quantification gives "for all $x$, $x^2 = 4$" (false). Closing by existential quantification gives "there exists $x$ such that $x^2 = 4$" (true, with $x = 2$ or $x = -2$).
Key Insight
The distinction between open and closed formulas is fundamental to predicate logic and model theory. The notion of a "free variable" is what separates algebraic equations from logical propositions.