note
Quantifiers & Implications Cheat Sheet
Updated 3/27/2026 • Resource: How to prove it?
| Concept | Symbolic Form | Negation | English Interpretation | Notes |
|---|---|---|---|---|
| Universal Quantifier | (\forall x \in D, P(x)) | (\exists x \in D, \neg P(x)) | “For all x in D, P(x) holds” | Flip to ∃ for negation |
| Existential Quantifier | (\exists x \in D, P(x)) | (\forall x \in D, \neg P(x)) | “There exists x in D such that P(x) holds” | Flip to ∀ for negation |
| Implication | (p \to q) | (p \land \neg q) | “If p then q” | False only when p=T, q=F |
| Negated Implication | (\neg(p \to q)) | (p \land \neg q) | “p is true and q is false” | Often used inside quantified statements |
| Nested Quantifiers | (\forall x \in D, \exists y \in D, P(x, y)) | (\exists x \in D, \forall y \in D, \neg P(x, y)) | “For every x, there exists y such that P(x, y)” | Order matters! |
| Vacuous Truth | (p \to q) when (p) is false | Always True | “If p then q is true whenever p is false” | Important in proofs |