Statements, Predicates & Quantifiers
Big Idea: In mathematics, we need precise language to express truths. Statements, predicates, and quantifiers are the building blocks of mathematical logic — the grammar of math itself.
1. Statements (Propositions)
A statement (or proposition) is a declarative sentence that is either TRUE or FALSE, but not both.
| Sentence | Statement? | Truth Value |
|---|---|---|
| $5 + 3 = 8$ | ✅ Yes | TRUE |
| $7 > 10$ | ✅ Yes | FALSE |
| "What time is it?" | ❌ No | — |
| "Close the door!" | ❌ No | — |
| $x + 2 = 5$ | ❌ Not yet* | Depends on $x$ |
*This is an open sentence — it becomes a statement only when we assign a value to $x$.
2. Predicates (Open Sentences)
A predicate is a sentence containing one or more variables that becomes a statement when specific values are substituted for those variables.
We write a predicate as $P(x)$, $Q(x, y)$, etc.
Example:
Let $P(x): x + 3 = 7$
• $P(4): 4 + 3 = 7$ → TRUE
• $P(1): 1 + 3 = 7$ → FALSE
• $P(0): 0 + 3 = 7$ → FALSE
Let $P(x): x + 3 = 7$
• $P(4): 4 + 3 = 7$ → TRUE
• $P(1): 1 + 3 = 7$ → FALSE
• $P(0): 0 + 3 = 7$ → FALSE
The truth set of $P(x)$ is the set of all values that make $P(x)$ true:
$$\{x \in U : P(x) \text{ is true}\}$$
where $U$ is the universal set (domain).