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Exercise (Recognizing Statements And Open Sentences)
Classify each sentence below as a statement, an open sentence, or neither. Make sure you justify each answer.
- Every integer is either even or odd.
- Today is Thursday.
- $x^2-9=0.$
- The second coming will occur in 2050.
- Sunsets are beautiful.
- Have you read the first book in the Harry Potter series?
- $\cos^2(x)+\sin^2(x)=1$.
Click to see a solution.
- The sentence "Every integer is either even or odd" is a statement, as it has the truth value of "True."
- The sentence "Today is Thursday" is an open sentence. The truth value depends on what the variable "today" is.
- The sentence "$x^2-9=0$" is an open sentence. The variable is $x$, and the value $x$ determines whether or not $x^2-9$ equals zero or not.
- The sentence "The second coming will occur in 2050" is a statement. It is either true or false, however we do not have the ability to determine the truth value (as we cannot see the future). The sentence is definitely either true or false, and not both, so it is a statement.
- The sentence "Sunsets are beautiful" is an opinion, and hence is neither a statement nor an open sentence.
- While "$\cos^2(x)+\sin^2(x)=1$" has a variable $x$ in it, we could argue that this is a statement and not an open sentence as the sentence is true for any value $x$ that makes sense in this sentence. However, since the variable $x$ was not specified, you could also argue that this is an open sentence. The context in which sentence occurs may alter whether a sentence is an open sentence or statement. If the sentence had instead read "For any real number $x$ we have $\cos^2(x)+\sin^2(x)=1$," then the sentence is definitely a true statement without any possible argument.