137 lines
4.5 KiB
Markdown
137 lines
4.5 KiB
Markdown
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---
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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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* **Existential quantification** asserts the existence of a member in a set (denoted the **range**) satisfying a property. There may be multiple members that satisfy the property; so long as one does, the existential quantification is considered true.
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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 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 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** 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 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** asserts that a number of 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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## Reference
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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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