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Integer encodings and discrete math.

c-declarations
Joshua Potter 2024-02-17 12:49:56 -07:00
parent 5fdd0b718e
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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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---
title: "2024-02-17"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Began working through two's-complement and unsigned encoding.
* Also added more notes on propositional logic.
* Taking a step back from "Concrete Mathematics" to first read through "Discrete Mathematics: An Open Introduction".

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@ -7,7 +7,7 @@ title: "2024-02-16"
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [x] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Read "Queries, Modeling, and Transformation" of "Fundamentals of Data Engineering".

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---
title: Binary Search
TARGET DECK: Obsidian::STEM
FILE TAGS: algorithm
tags:
- algorithm
---
## Overview
Property | Value
----------- | --------
Best Case | $O(1)$
Worst Case | $O(\lg{n})$
Aux. Memory | $O(1)$
%%ANKI
Basic
What is the best case running time of binary search?
Back: $\Omega(1)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310004-->
END%%
%%ANKI
Basic
What input does binary search perform best on?
Back: One in which the value being searched for is already in the middle.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310011-->
END%%
%%ANKI
Basic
What is the worst case running time of binary search?
Back: $O(\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310015-->
END%%
%%ANKI
Basic
What input does binary search perform worst on?
Back: One in which the value does not exist.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310018-->
END%%
%%ANKI
Basic
What is the typical output of binary search?
Back: The index of the element in the array being searched for, if found.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310021-->
END%%
A recursive solution looks as follows:
```c
static int aux(const int needle, const int i, const int j, int *A) {
if (i > j) {
return -1;
}
int mid = (i + j) / 2;
if (A[mid] == needle) {
return mid;
} else if (A[mid] < needle) {
return aux(needle, mid + 1, j, A);
} else {
return aux(needle, i, mid - 1, A);
}
}
int binary_search(const int needle, const int n, int A[static n]) {
return aux(needle, 0, n - 1, A);
}
```
We can also write this iteratively:
```c
int binary_search(const int needle, const int n, int A[static n]) {
int i = 0;
int j = n - 1;
while (i <= j) {
int mid = (i + j) / 2;
if (A[mid] == needle) {
return mid;
} else if (A[mid] < needle) {
i = mid + 1;
} else {
j = mid - 1;
}
}
return -1;
}
```
%%ANKI
Basic
In binary search, when could using floor for midpoint calculations yield different answers than ceiling?
Back: When there exist duplicate members.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708174545522-->
END%%
%%ANKI
Basic
In binary search, what ensures left pointer `L` and right pointers `R` eventually satisfy `L > R`?
Back: The found midpoint is always excluded from the next binary search invocation.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708174545527-->
END%%
## References
* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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---
title: Integer Encoding
TARGET DECK: Obsidian::STEM
FILE TAGS: binary
tags:
- binary
---
## Overview
Integers are typically encoded using either **unsigned encoding** or **two's complement**.
%%ANKI
Basic
What is a C integral type?
Back: A type representing finite ranges of integers.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1708177246087-->
END%%
%%ANKI
Basic
In what two ways are C integral types encoded?
Back: Unsigned encoding or two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1708177246093-->
END%%
%%ANKI
Basic
What integer values does unsigned encoding represent?
Back: Nonnegative values.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246097-->
END%%
%%ANKI
Basic
What integer values does two's-complement represent?
Back: Negative, zero, and positive values.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246102-->
END%%
%%ANKI
Basic
An integral value of 0 has what encoding?
Back: Either unsigned or two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246105-->
END%%
%%ANKI
Basic
An integral value of 100 has what encoding?
Back: Either unsigned or two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246109-->
END%%
%%ANKI
Basic
An integral value of -100 has what encoding?
Back: Two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246114-->
END%%
### Unsigned Encoding
Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$$
%%ANKI
Basic
What is the largest unsigned value an integral type of $w$ bits can encode?
Back: $2^w - 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246119-->
END%%
%%ANKI
Basic
What is the smallest unsigned value an integral type of $w$ bits can encode?
Back: $0$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246123-->
END%%
%%ANKI
Basic
What half-open interval represents the possible $w$-bit unsigned values?
Back: $[0, 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246128-->
END%%
%%ANKI
Basic
What is the binary representation of the smallest $4$-bit unsigned number?
Back: $0000_2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246133-->
END%%
%%ANKI
Basic
What is the binary representation of the largest $4$-bit unsigned number?
Back: $1111_2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246138-->
END%%
%%ANKI
Basic
What is the decimal expansion of unsigned type $1010_2$?
Back: $2^3 + 2^1 = 10$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246143-->
END%%
%%ANKI
Basic
What does the "uniqueness" of unsigned encoding refer to?
Back: The function used to convert integral types to their unsigned encoding is a bijection.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246148-->
END%%
%%ANKI
Basic
How does Bryant et al. define $B2U_w$?
Back: $B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147785-->
END%%
%%ANKI
Basic
What is $B2U_w$ an acronym for?
Back: **B**inary "to" **u**nsigned.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147791-->
END%%
%%ANKI
Basic
What does $w$ in $B2U_w$ represent?
Back: The number of bits in the integral type being interpreted.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147795-->
END%%
%%ANKI
Basic
What is the domain of $B2U_w$?
Back: Bit vectors of size $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147798-->
END%%
%%ANKI
Basic
What is the range of $B2U_w$?
Back: $[0, 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147801-->
END%%
%%ANKI
Basic
Why is "$B2U$" insufficient for use?
Back: We need to understand how many bits conversion is with respect to.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147804-->
END%%
### Two's-Complement
Represents negative numbers along with nonnegative ones. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its twos'-complement encoding with: $$B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$$
%%ANKI
Basic
What is the largest two's-complement value an integral type of $w$ bits can encode?
Back: $2^{w-1} - 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147807-->
END%%
%%ANKI
Basic
What is the smallest two's-complement value an integral type of $w$ bits can encode?
Back: $-2^{w-1}$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147810-->
END%%
%%ANKI
Basic
What half-open interval represents the possible $w$-bit two's-complement values?
Back: $[-2^{w-1}, 2^{w-1})$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246128-->
END%%
%%ANKI
Cloze
$[${1:$0$}, {2:$2^w$}$)$ is to unsigned as $[${1:$-2^{w-1}$}, {2:$2^{w-1}$}$)$ is to two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147813-->
END%%
%%ANKI
Basic
What is the binary representation of the smallest $4$-bit two's-complement number?
Back: $1000_2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649872-->
END%%
%%ANKI
Basic
What is the binary representation of the largest $4$-bit two's-complement number?
Back: $0111_2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649876-->
END%%
%%ANKI
Cloze
The {sign bit} refers to the {most significant bit} in two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649881-->
END%%
%%ANKI
Basic
What is the weight of the sign bit in $w$-bit two's-complement?
Back: $-2^{w-1}$
The {sign bit} refers to the {most significant bit} in two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649887-->
END%%
%%ANKI
Basic
What does the "uniqueness" of two's-complement refer to?
Back: The function used to convert integral types to two's-complement is a bijection.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649894-->
END%%
%%ANKI
Basic
How does Bryant et al. define $B2T_w$?
Back: $B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649901-->
END%%
%%ANKI
Basic
What is $B2T_w$ an acronym for?
Back: **B**inary "to" **t**wo's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649907-->
END%%
%%ANKI
Basic
What does $w$ in $B2T_w$ represent?
Back: The number of bits in the integral type being interpreted.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649913-->
END%%
%%ANKI
Basic
What is the domain of $B2T_w$?
Back: Bit vectors of size $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649921-->
END%%
%%ANKI
Basic
What is the range of $B2T_w$?
Back: $[-2^{w-1}, 2^{w-1})$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649928-->
END%%
%%ANKI
Basic
Why is "$B2T$" insufficient for use?
Back: We need to understand how many bits conversion is with respect to.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649935-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

26
notes/logic/equiv-trans.md
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@ -9,7 +9,7 @@ tags:
## Overview
**Equivalence-transformation** refers to a class of calculi for manipulating propositions 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.
**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.
%%ANKI
Basic
@ -27,14 +27,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
<!--ID: 1707422675517-->
END%%
%%ANKI
Basic
What are the basic propositional logical operators?
Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$/$=$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861291-->
END%%
%%ANKI
Cloze
Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
@ -72,22 +64,6 @@ Tags: lean
<!--ID: 1706994861302-->
END%%
%%ANKI
Basic
What name is given to operand $a$ in $a \Rightarrow b$?
Back: The antecedent
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861308-->
END%%
%%ANKI
Basic
What name is given to operand $b$ in $a \Rightarrow b$?
Back: The consequent
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861310-->
END%%
%%ANKI
Basic
Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?

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---
title: Propositional Logic
TARGET DECK: Obsidian::STEM
FILE TAGS: logic::propositional
tags:
- logic
- propositional
---
## Overview
A branch of logic derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and biconditionals ($\Leftrightarrow$). There exists a hierarchy of terms used to describe a string of English:
* A **sentence** is any grammatical string of words.
* A **predicate** is a sentence with free variables.
* A **statement** is a sentence that can be assigned a truth or false value.
* A predicate with free variables "plugged in" is a statement.
%%ANKI
Basic
What are the basic propositional logical operators?
Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861291-->
END%%
%%ANKI
Basic
What is a propositional statement?
Back: A declarative sentence which is either true or false.
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).
<!--ID: 1708199272076-->
END%%
%%ANKI
Basic
What two categories do propositional statements fall within?
Back: Atomic and molecular statements.
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).
<!--ID: 1708199272083-->
END%%
%%ANKI
Basic
What is an atomic statement?
Back: It cannot be broken up into smaller statements.
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).
<!--ID: 1708199272087-->
END%%
%%ANKI
Basic
What is a molecular statement?
Back: It can be broken up into smaller statements.
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).
<!--ID: 1708199272091-->
END%%
%%ANKI
Cloze
A {molecular} statement can be broken up into {atomic} statements.
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).
<!--ID: 1708199272095-->
END%%
%%ANKI
Basic
What distinguishes a sentence from a statement?
Back: The latter is a sentence that can be derived a truth value.
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).
<!--ID: 1708199272099-->
END%%
%%ANKI
Basic
What distinguishes a sentence from a predicate?
Back: The latter is a sentence that contains free variables.
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).
<!--ID: 1708199272103-->
END%%
%%ANKI
Basic
What distinguishes a predicate from a statement?
Back: A statement does not contain free variables.
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).
<!--ID: 1708199272110-->
END%%
%%ANKI
Basic
How are statements defined in terms of predicates?
Back: A statement is a predicate with $0$ free variables.
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).
<!--ID: 1708199272115-->
END%%
%%ANKI
Basic
Why is "$3 + x = 12$" *not* a statement?
Back: Because $x$ is a variable.
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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END%%
## Implication
Implication is denoted as $\Rightarrow$. It has truth table
$p$ | $q$ | $p \Rightarrow q$
--- | --- | -----------------
$T$ | $T$ | $T$
$T$ | $F$ | $F$
$F$ | $T$ | $T$
$F$ | $F$ | $T$
Implication has a few "equivalent" English expressions that are commonly used.
Given propositions $P$ and $Q$, we have the following equivalences:
* $P$ if $Q$
* $P$ only if $Q$
* $P$ is necessary for $Q$
* $P$ is sufficient for $Q$
%%ANKI
Basic
What name is given to operand $a$ in $a \Rightarrow b$?
Back: The antecedent
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861308-->
END%%
%%ANKI
Basic
What name is given to operand $b$ in $a \Rightarrow b$?
Back: The consequent
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861310-->
END%%
%%ANKI
Basic
How does "$P$ if $Q$" translate with $\Rightarrow$?
Back: $Q \Rightarrow P$
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).
<!--ID: 1708199272127-->
END%%
%%ANKI
Basic
How does "$P$ only if $Q$" translate with $\Rightarrow$?
Back: $P \Rightarrow Q$
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).
<!--ID: 1708199272134-->
END%%
%%ANKI
Basic
How does "$P$ is necessary for $Q$" translate with $\Rightarrow$?
Back: $Q \Rightarrow P$
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).
<!--ID: 1708199272140-->
END%%
%%ANKI
Basic
How does "$P$ is sufficient for $Q$" translate with $\Rightarrow$?
Back: $P \Rightarrow Q$
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).
<!--ID: 1708199272145-->
END%%
%%ANKI
Basic
Which of *if*/*only if* map to *necessary*?
Back: *if*
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).
<!--ID: 1708199272151-->
END%%
%%ANKI
Basic
Which of *if*/*only if* map to *sufficient*?
Back: *only if*
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).
<!--ID: 1708199272157-->
END%%
%%ANKI
Basic
Which logical operator maps to "if and only if"?
Back: $\Leftrightarrow$
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).
<!--ID: 1708199272163-->
END%%
%%ANKI
Basic
Which logical operator maps to "necessary and sufficient"?
Back: $\Leftrightarrow$
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).
<!--ID: 1708199272168-->
END%%
%%ANKI
Basic
What is the converse of $P \Rightarrow Q$?
Back: $Q \Rightarrow P$
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).
<!--ID: 1708199272173-->
END%%
%%ANKI
Basic
When is implication equivalent to its converse?
Back: It's indeterminate.
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).
<!--ID: 1708199272178-->
END%%
%%ANKI
Basic
What is the contrapositive of $P \Rightarrow Q$?
Back: $\neg Q \Rightarrow \neg P$
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).
<!--ID: 1708199272184-->
END%%
%%ANKI
Basic
When is implication equivalent to its contrapositive?
Back: They are always equivalent.
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).
<!--ID: 1708199272189-->
END%%
## References
* 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).

11
notes/logic/quantification.md
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@ -53,6 +53,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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END%%
%%ANKI
Basic
What term refers to $S$ in $\exists x : S, P(x)$?
Back: The domain of discourse.
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).
<!--ID: 1708199272194-->
END%%
%%ANKI
Basic
What is the identity element of $\lor$?
@ -227,4 +235,5 @@ END%%
## References
* 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).