547 lines
18 KiB
Markdown
547 lines
18 KiB
Markdown
---
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title: Equivalence Transformation
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TARGET DECK: Obsidian::STEM
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FILE TAGS: programming::equiv-trans
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tags:
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- equiv-trans
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- logic
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- programming
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---
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## Overview
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**Equivalence-transformation** refers to a class of calculi for [[propositional|propositional logic]] derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$). Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
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%%ANKI
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Basic
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Who is the author of "The Science of Programming"?
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Back: David Gries
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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: 1706994861286-->
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END%%
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%%ANKI
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Basic
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What are constant propositions?
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Back: Propositions that contain only constants as operands.
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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: 1707422675517-->
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END%%
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%%ANKI
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Cloze
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Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
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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: 1706994861295-->
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END%%
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%%ANKI
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Basic
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How does Lean define propositional equality?
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Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
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Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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Tags: lean
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<!--ID: 1706994861298-->
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END%%
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%%ANKI
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Basic
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How does Lean define `propext`?
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Back:
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```lean
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axiom propext {a b : Prop} : (a ↔ b) → (a = b)
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```
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Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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Tags: lean
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<!--ID: 1706994861300-->
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END%%
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%%ANKI
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Basic
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What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
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Back: `propext`
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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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Tags: lean
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<!--ID: 1706994861302-->
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END%%
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%%ANKI
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Basic
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Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
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Back: Yes.
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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: 1706994861318-->
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END%%
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%%ANKI
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Basic
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Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
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Back: No.
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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: 1706994861320-->
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END%%
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%%ANKI
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Basic
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What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
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Back: $b \lor \neg c$
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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: 1706994861337-->
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END%%
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%%ANKI
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Basic
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What set of states does $a \land b$ represent?
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Back: The set containing just state $\{(a, T), (b, T)\}$.
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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: 1706994861339-->
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END%%
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%%ANKI
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Basic
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What is sloppy about phrase "the states in $b \lor \neg c$"?
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Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
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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: 1706994861341-->
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END%%
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%%ANKI
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Basic
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What is the weakest proposition?
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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: 1706994861348-->
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END%%
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%%ANKI
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Basic
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What set of states does $T$ represent?
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Back: The set of all states.
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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: 1706994861350-->
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END%%
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%%ANKI
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Basic
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What is the strongest proposition?
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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: 1706994861352-->
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END%%
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%%ANKI
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Basic
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What set of states does $F$ represent?
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Back: The set of no states.
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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: 1706994861354-->
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END%%
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%%ANKI
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Basic
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What does a proposition *represent*?
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Back: The set of states in which it is true.
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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: 1706994861335-->
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END%%
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%%ANKI
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Basic
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When is $p$ stronger than $q$?
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Back: When $p \Rightarrow q$.
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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: 1706994861343-->
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END%%
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%%ANKI
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Basic
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When is $p$ weaker than $q$?
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Back: When $q \Rightarrow p$.
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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: 1706994861346-->
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END%%
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%%ANKI
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Basic
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A proposition is well-defined with respect to what?
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Back: A state to evaluate against.
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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: 1706994861316-->
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END%%
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%%ANKI
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Basic
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Why is $b \land c$ stronger than $b \lor c$?
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Back: The former represents a subset of the states the latter represents.
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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: 1706994861356-->
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END%%
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%%ANKI
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Basic
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What is a state?
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Back: A function mapping identifiers to values.
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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: 1706994861314-->
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END%%
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%%ANKI
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Basic
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What are the two calculi Gries describes equivalence-transformation with?
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Back: A formal system and a system of evaluation.
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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: 1707251673342-->
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END%%
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## Equivalence Schemas
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A proposition is said to be a **tautology** if it evaluates to $T$ in every state it is well-defined in. We say propositions $E1$ and $E2$ are **equivalent** if $E1 = E2$ is a tautology. In this case, we say $E1 = E2$ is an **equivalence**.
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%%ANKI
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Basic
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What does it mean for a proposition to be a tautology?
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Back: That the proposition is true in every state it is well-defined in.
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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: 1706994861323-->
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END%%
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%%ANKI
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Basic
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How is tautology $e$ written equivalently with a quantifier?
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Back: For free identifiers $i_1, \ldots, i_n$ in $e$, as $\forall (i_1, \ldots, i_n), e$.
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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: 1707937867032-->
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END%%
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%%ANKI
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Basic
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The term "equivalent" refers to a comparison between what two objects?
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Back: Expressions.
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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: 1707251673345-->
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END%%
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%%ANKI
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Basic
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What does it mean for two propositions to be equivalent?
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Back: Given propositions $E1$ and $E2$, it means $E1 = E2$ is a tautology.
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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: 1707251673347-->
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END%%
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%%ANKI
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Basic
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What is an equivalence?
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Back: Given propositions $E1$ and $E2$, tautology $E1 = E2$.
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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: 1707251673348-->
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END%%
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* Commutative Laws
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* $(E1 \land E2) = (E2 \land E1)$
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* $(E1 \lor E2) = (E2 \lor E1)$
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* $(E1 = E2) = (E2 = E1)$
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%%ANKI
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Basic
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Which of the basic logical operators do the commutative laws apply to?
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Back: $\land$, $\lor$, and $=$
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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: 1707251673350-->
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END%%
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%%ANKI
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Basic
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What do the commutative laws allow us to do?
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Back: Reorder operands.
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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: 1707251673351-->
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END%%
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%%ANKI
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Basic
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What is the commutative law of e.g. $\land$?
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Back: $E1 \land E2 = E2 \land E1$
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<!--ID: 1707251673353-->
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END%%
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* Associative Laws
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* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
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* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$
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%%ANKI
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Basic
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Which of the basic logical operators do the associative laws apply to?
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Back: $\land$ and $\lor$
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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: 1707251673354-->
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END%%
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%%ANKI
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Basic
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What do the associative laws allow us to do?
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Back: Remove parentheses.
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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: 1707251673355-->
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END%%
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%%ANKI
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Basic
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What is the associative law of e.g. $\land$?
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Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
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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: 1707251673357-->
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END%%
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* Distributive Laws
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* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$
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* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
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%%ANKI
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Basic
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Which of the basic logical operators do the distributive laws apply to?
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Back: $\land$ and $\lor$
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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: 1707251673358-->
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END%%
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%%ANKI
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Basic
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What do the distributive laws allow us to do?
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Back: "Factor" propositions.
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Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673360-->
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END%%
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%%ANKI
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Basic
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What is the distributive law of e.g. $\land$ over $\lor$?
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Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
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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: 1707251673361-->
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END%%
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* De Morgan's Laws
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* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
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* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$
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%%ANKI
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Basic
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Which of the basic logical operators do De Morgan's Laws apply to?
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Back: $\neg$, $\land$, and $\lor$
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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: 1707251673363-->
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END%%
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%%ANKI
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Basic
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What is De Morgan's Law of e.g. $\land$?
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Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
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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: 1707251673364-->
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END%%
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* Law of Negation
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* $\neg (\neg E1) = E1$
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%%ANKI
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Basic
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What does the Law of Negation say?
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Back: $\neg (\neg E1) = E1$
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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: 1707251673365-->
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END%%
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* Law of the Excluded Middle
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* $E1 \lor \neg E1 = T$
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%%ANKI
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Basic
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Which of the basic logical operators does the Law of the Excluded Middle apply to?
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Back: $\lor$
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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: 1707251673367-->
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END%%
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%%ANKI
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Basic
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What does the Law of the Excluded Middle say?
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Back: $E1 \lor \neg E1 = T$
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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: 1707251673368-->
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END%%
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%%ANKI
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Basic
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Which equivalence schema is "refuted" by sentence, "This sentence is false."
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Back: Law of the Excluded Middle
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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: 1707251779153-->
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END%%
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* Law of Contradiction
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* $E1 \land \neg E1 = F$
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%%ANKI
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Basic
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Which of the basic logical operators does the Law of Contradiction apply to?
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Back: $\land$
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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: 1707251673370-->
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END%%
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%%ANKI
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Basic
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What does the Law of Contradiction say?
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Back: $E1 \land \neg E1 = F$
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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: 1707251673371-->
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END%%
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%%ANKI
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Cloze
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The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} 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: 1707251673373-->
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END%%
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Gries lists other "Laws" but they don't seem as important to note here.
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%%ANKI
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Basic
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How is $\Rightarrow$ written in terms of other logical operators?
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Back: $p \Rightarrow q$ is equivalent to $\neg p \lor q$.
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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: 1706994861358-->
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END%%
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%%ANKI
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Basic
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How is $\Leftrightarrow$/$=$ written in terms of other logical operators?
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Back: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$.
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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: 1706994861360-->
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END%%
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%%ANKI
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Basic
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What distinguishes an equality from an equivalence?
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Back: An equivalence is an equality that is also a tautology.
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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: 1707316178709-->
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END%%
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## Equivalence Rules
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* Rule of Substitution
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* Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
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* Rule of Transitivity
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* Let $E1 = E2$ and $E2 = E3$ be equivalences. Then $E1 = E3$ is an equivalence.
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%%ANKI
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Basic
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What two inference rules make up the equivalence-transformation formal system?
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Back: Substitution and transitivity.
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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: 1707253246450-->
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END%%
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%%ANKI
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Basic
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Which of the two inference rules that make up the equivalence-transformation formal system is redundant?
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Back: Transitivity.
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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: 1707432641598-->
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END%%
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%%ANKI
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Basic
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What does the rule of substitution say in the system of evaluation?
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Back: Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
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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: 1707253246452-->
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END%%
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%%ANKI
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Basic
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How is the rule of substitution written as an inference rule (in standard form)?
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Back:
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$$
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\begin{matrix}
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E1 = E2 \\
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\hline P(E1) = P(E2)
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\end{matrix}
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$$
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<!--ID: 1707253246454-->
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END%%
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%%ANKI
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Basic
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What does the rule of transitivity state in the system of evaluation?
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Back: Let $E1 = E2$ and $E2 = E3$. Then $E1 = E3$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707253246455-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is the rule of transitivity written as an inference rule (in standard form)?
|
||
Back:
|
||
$$
|
||
\begin{matrix}
|
||
E1 = E2, E2 = E3 \\
|
||
\hline E1 = E3
|
||
\end{matrix}
|
||
$$
|
||
<!--ID: 1707253246457-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
The system of evaluation has {equivalences} whereas the formal system has {theorems}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707253246458-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is a "theorem" in the equivalence-transformation formal system?
|
||
Back: An equivalence derived from the axioms and inference rules.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707316178712-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is e.g. the Law of Implication proven in the system of evaluation?
|
||
Back: With truth tables
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707316178714-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is e.g. the Law of Implication proven in the formal system?
|
||
Back: It isn't. It is an axiom.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707316178715-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
The system of evaluation and formal system are connected by the following biconditional: {$e$ is a tautology} iff {$e = T$ is a theorem}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707316178717-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
The {1:system of evaluation} is to {2:"$e$ is a tautology"} whereas the {2:formal system} is to {1:"$e = T$ is a theorem"}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707316276203-->
|
||
END%%
|
||
|
||
## Bibliography
|
||
|
||
* Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
|
||
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. |