Recognizing atomic and compound statements

Statement

For each sentence, decide whether it is an atomic statement, a compound statement, or not a statement.

1. The number $12$ is even.
2. Close the door!
3. If it rains, then I take the umbrella.
4. $7$ is odd and $4$ is a prime number.
5. $x + 2 = 9$.
6. This sentence is false.

Solution

1. Atomic statement: it expresses a single idea that is true or false (here it is true).
2. Not a statement: it is a command, with no truth value.
3. Compound statement: it uses the connective if… then (implication).
4. Compound statement: it uses the connective and (conjunction); it is false because $4$ is not prime.
5. Not a statement: it contains the free variable $x$; it becomes a statement only by assigning a value to $x$.
6. Not a statement: it is self-referential (the liar paradox), so it can be neither true nor false.