A **quantifier** refers to an operator that specifies how many members of a set satisfy some formula. The most common quantifiers are $\exists$ and $\forall$, though others (such as the counting quantifier) are also used.
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Basic
What are the most common first-order logic quantifiers?
Back: $\exists$ and $\forall$
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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%%ANKI
Basic
What term refers to operators like $\exists$ and $\forall$?
Back: Quantifiers.
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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* **Existential quantification** ($\exists$) asserts the existence of at least one member in a set satisfying a property.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free**. A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix**), and the second consisting of no quantifiers (the **matrix**).
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Basic
Prenex normal form consists of what two parts?
Back: The prefix and the matrix.
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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Basic
How is the prefix of a formula in PNF formatted?
Back: As only quantifiers and bound variables.
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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Basic
How is the matrix of a formula in PNF formatted?
Back: Without quantifiers.
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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Basic
Which identifiers in the following are bound? $$\exists x, P(x) \land P(y)$$
Back: Just $x$.
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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%%ANKI
Basic
Which identifiers in the following are free? $$\exists x, P(x) \land P(y)$$
Back: Just $y$.
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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%%ANKI
Basic
How is the following rewritten in PNF? $$(\exists x, P(x)) \land (\exists y, P(y))$$
Back: $\exists x \;y, P(x) \land P(y)$
Reference: Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
* Gries, David.*The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).