Here's a breakdown:
* Variable: A symbol that represents an unknown value. It can be a letter (like 'x' or 'y') or other symbols.
* Truth Value: Whether a statement is true or false.
* Open Sentence: A sentence that includes a variable, making its truth value uncertain until the variable is replaced with a specific value.
Examples:
* "x + 2 = 5" is an open sentence. Its truth value depends on the value of 'x'. If 'x' is 3, then the sentence is true. If 'x' is any other value, the sentence is false.
* "The dog is brown." This is not an open sentence because it doesn't contain a variable. Its truth value is determined by the dog's actual color.
Key Characteristics of Open Sentences:
* Can't be determined as true or false without assigning values to variables.
* Often used in algebra and logic to represent relationships and solve equations.
In contrast to open sentences, a closed sentence is a statement whose truth value is fixed and can be determined without knowing the value of any variable.
Examples of closed sentences:
* "The sky is blue." This sentence is either true or false, regardless of any variables.
* "2 + 2 = 4." This is a mathematical statement that is always true.