Operations on Statements and Predicates
What is a Statement (Proposition)?
A statement (or proposition) is a declarative sentence that is either TRUE (T) or FALSE (F), but never both.
✓ Statements
- "5 is a prime number." → TRUE
- "$2 + 3 = 6$" → FALSE
- "Every square is a rectangle." → TRUE
- "$\sqrt{4} = 3$" → FALSE
✗ NOT Statements
- "What time is it?" (question)
- "Close the door!" (command)
- "$x + 5 = 10$" (open sentence — depends on $x$)
- "This sentence is false." (paradox)
What is a Predicate?
A predicate is an open sentence containing one or more variables. It becomes a statement only when the variables are replaced by specific values from a given domain.
Notation: We write $P(x)$ to denote a predicate $P$ with variable $x$.
For example, $P(x): x + 3 = 7$ is a predicate. When $x = 4$, $P(4): 4 + 3 = 7$ is TRUE. When $x = 2$, $P(2): 2 + 3 = 7$ is FALSE.
For example, $P(x): x + 3 = 7$ is a predicate. When $x = 4$, $P(4): 4 + 3 = 7$ is TRUE. When $x = 2$, $P(2): 2 + 3 = 7$ is FALSE.
The set of all values that make the predicate true is called the truth set (or solution set).