239 lines
8.5 KiB
Markdown
239 lines
8.5 KiB
Markdown
---
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title: Quantification
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TARGET DECK: Obsidian::STEM
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FILE TAGS: logic::quantification
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tags:
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- logic
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- quantification
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---
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## Overview
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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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%%ANKI
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Basic
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What are the most common first-order logic quantifiers?
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Back: $\exists$ and $\forall$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796763-->
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END%%
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%%ANKI
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Basic
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What term refers to operators like $\exists$ and $\forall$?
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Back: Quantifiers.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796766-->
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END%%
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* **Existential quantification** ($\exists$) asserts the existence of at least one member in a set satisfying a property.
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%%ANKI
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Basic
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What symbol denotes existential quantification?
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Back: $\exists$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819964-->
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END%%
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%%ANKI
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Basic
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How many members in the domain of discourse must satisfy a property in existential quantification?
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Back: At least one.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819967-->
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END%%
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%%ANKI
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Basic
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$\exists x : S, P(x)$ is shorthand for what?
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Back: $\exists x, x \in S \land P(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819968-->
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END%%
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%%ANKI
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Basic
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What term refers to $S$ in $\exists x : S, P(x)$?
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Back: The domain of discourse.
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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).
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<!--ID: 1708199272194-->
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END%%
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%%ANKI
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Basic
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What is the identity element of $\lor$?
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Back: $F$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819970-->
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END%%
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* **Universal quantification** ($\forall$) asserts that every member of a set satisfies a property.
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%%ANKI
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Basic
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What symbol denotes universal quantification?
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Back: $\forall$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819971-->
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END%%
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%%ANKI
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Basic
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How many members in the domain of discourse must satisfy a property in universal quantification?
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Back: All of them.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819973-->
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END%%
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%%ANKI
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Basic
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$\forall x : S, P(x)$ is shorthand for what?
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Back: $\forall x, x \in S \Rightarrow P(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819976-->
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END%%
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%%ANKI
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Basic
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What is the identity element of $\land$?
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Back: $T$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819978-->
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END%%
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%%ANKI
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Cloze
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{1:$\exists$} is to {2:$\lor$} as {2:$\forall$} is to {1:$\land$}.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819979-->
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END%%
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%%ANKI
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Basic
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How is $\forall x : S, P(x)$ equivalently written in terms of existential quantification?
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Back: $\neg \exists x : S, \neg P(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819981-->
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END%%
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%%ANKI
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How is $\exists x : S, P(x)$ equivalently written in terms of universal quantification?
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Back: $\neg \forall x : S, \neg P(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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* **Counting quantification** ($\exists^{=k}$ or $\exists^{\geq k}$) asserts that (at least) $k$ (say) members of a set satisfy a property.
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%%ANKI
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Basic
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What symbol denotes counting quantification (of *exactly* $k$ members)?
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Back: $\exists^{=k}$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819983-->
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END%%
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%%ANKI
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Basic
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What symbol denotes counting quantification (of *at least* $k$ members)?
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Back: $\exists^{\geq k}$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494819985-->
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END%%
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%%ANKI
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Basic
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How is $\exists x : S, P(x)$ written in terms of counting quantification?
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Back: $\exists^{\geq 1} x : S, P(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494832056-->
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END%%
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%%ANKI
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Basic
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How is $\forall x : S, P(x)$ written in terms of counting quantification?
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Back: Assuming $S$ has $k$ members, $\exists^{= k} x : S, P(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707494832058-->
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END%%
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## Identifiers
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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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%%ANKI
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Basic
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When is an identifier said to be bound?
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Back: When it is specified as a parameter to a quantifier.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796768-->
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END%%
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%%ANKI
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Basic
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When is an identifier said to be free?
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Back: When it isn't specified as a parameter to a quantifier.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796770-->
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END%%
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%%ANKI
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Cloze
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An identifier that is not {bound} is instead {free}.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796772-->
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END%%
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%%ANKI
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Basic
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Prenex normal form consists of what two parts?
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Back: The prefix and the matrix.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796773-->
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END%%
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%%ANKI
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Basic
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How is the prefix of a formula in PNF formatted?
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Back: As only quantifiers and bound variables.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796775-->
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END%%
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%%ANKI
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Basic
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How is the matrix of a formula in PNF formatted?
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Back: Without quantifiers.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796776-->
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END%%
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%%ANKI
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Basic
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Which identifiers in the following are bound? $$\exists x, P(x) \land P(y)$$
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Back: Just $x$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796777-->
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END%%
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%%ANKI
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Basic
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Which identifiers in the following are free? $$\exists x, P(x) \land P(y)$$
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Back: Just $y$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707674796779-->
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END%%
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%%ANKI
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Basic
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How is the following rewritten in PNF? $$(\exists x, P(x)) \land (\exists y, P(y))$$
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Back: $\exists x \;y, P(x) \land P(y)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707675399517-->
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END%%
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## References
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* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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* 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). |