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c-declarations
Joshua Potter 2024-02-20 07:13:03 -07:00
parent 27adc88529
commit 8d6978c429
9 changed files with 312 additions and 41 deletions
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13
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notes/_journal/2024-02-20.md Normal file
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---
title: "2024-02-20"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [x] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Added `printf` `length` field notes.
* 101weiqi (serial numbers)
* Q-138432
* Q-266721
* Q-110886
* Q-267289
* Q-275961
* Q-324650
* Q-83832

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notes/_journal/2024-02-19.md → notes/_journal/2024-02/2024-02-19.md
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@ -4,7 +4,7 @@ title: "2024-02-19"
- [x] Anki Flashcards - [x] Anki Flashcards
- [x] KoL - [x] KoL
- [ ] Sheet Music (10 min.) - [x] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem) - [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story) - [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem) - [ ] Interview Prep (1 Practice Problem)

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notes/binary/index.md
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@ -8,7 +8,7 @@ tags:
## Overview ## Overview
A binary digit or **bit** is a `0` or `1` character. A bit string is then a contiguous sequence of bits. It's **weight** is a reference to the number of `1`s in the bit string. Compare the below operation to the method for converting from one numerical base to another (e.g. [[hexadecimal]]). A binary digit or **bit** is a `0` or `1` character. A **bit string** is then a contiguous sequence of bits. It's **weight** is a reference to the number of `1`s in the bit string. Compare the below operation to the method for converting from one numerical base to another (e.g. [[hexadecimal]]).
```c ```c
unsigned int bit_weight(int64_t n) { unsigned int bit_weight(int64_t n) {

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notes/binary/shifts.md
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@ -105,7 +105,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How is $x \bmod 2^k$ equivalently written as a bit mask? How is $x \bmod 2^k$ equivalently written as a bit mask?
Back: `x & (k - 1)` Back: `x & ((1 << k) - 1)`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c Tags: c
<!--ID: 1707873410780--> <!--ID: 1707873410780-->

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notes/c/declarations.md
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@ -8,15 +8,17 @@ tags:
## Overview ## Overview
Signed | Unsigned | 32-bit | 64-bit Signed | Unsigned | 32-bit | 64-bit
----------- | -------------- | ------ | ------ ----------- | ------------------- | ------ | ------
signed char | unsigned char | 1 | 1 signed char | unsigned char | 1 | 1
short | unsigned short | 2 | 2 short | unsigned short | 2 | 2
int | unsigned | 4 | 4 int | unsigned | 4 | 4
long | unsigned long | 4 | 8 long | unsigned long | 4 | 8
char * | - | 4 | 8 long long | unsigned long long | 8 | 8
float | - | 4 | 4 char * | - | 4 | 8
double | - | 8 | 8 float | - | 4 | 4
double | - | 8 | 8
Fixed width data integral types (e.g. `int32_t`) can be found by including `<stdint.h>`. Fixed width data integral types (e.g. `int32_t`) can be found by including `<stdint.h>`.
%%ANKI %%ANKI
@ -103,6 +105,29 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1707493017222--> <!--ID: 1707493017222-->
END%% END%%
%%ANKI
Cloze
`long long` *typically* represents {8} bytes(s) on a 64-bit platform.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708425521263-->
END%%
%%ANKI
Basic
What distinguishes `long` from `long long`?
Back: `long long`s are guaranteed to be at least 64-bit wide.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708425521292-->
END%%
%%ANKI
Basic
*Why* is there both a `long` and `long long`?
Back: `long long`s are at least 64-bit wide, even on 32-bit platforms.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708425521296-->
END%%
%%ANKI %%ANKI
Cloze Cloze
`char *` *typically* represents {8} bytes(s) on a 64-bit platform. `char *` *typically* represents {8} bytes(s) on a 64-bit platform.

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@ -61,6 +61,13 @@ Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia
<!--ID: 1708384441472--> <!--ID: 1708384441472-->
END%% END%%
%%ANKI
Cloze
The {`width` and `precision`} fields are output related whereas the {`length`} field is input related.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708425941269-->
END%%
Flag | Description Flag | Description
--------- | ----------- --------- | -----------
`-` | Left-aligns the output `-` | Left-aligns the output
@ -232,6 +239,57 @@ Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia
<!--ID: 1707918756888--> <!--ID: 1707918756888-->
END%% END%%
Length | Description
--------- | -----------
`hh` | `int` sized integer argument promoted from a `char`
`h` | `int` sized integer argument promoted from a `short`
`l` | `long` sized integer argument
`ll` | `long long` sized integer argument
`z` | `size_t` sized integer argument
%%ANKI
Basic
*Why* do length modifiers for e.g. `char` exist?
Back: For maximum portability in the face of default argument promotions.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708426673637-->
END%%
%%ANKI
Cloze
The {`hh`} length corresponds to the {`char`} type declaration.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708426673664-->
END%%
%%ANKI
Cloze
The {`h`} length corresponds to the {`short`} type declaration.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708426673668-->
END%%
%%ANKI
Cloze
The {`l`} length corresponds to the {`long`} type declaration.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708426673711-->
END%%
%%ANKI
Cloze
The {`ll`} length corresponds to the {`long long`} type declaration.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708426673719-->
END%%
%%ANKI
Cloze
The {`z`} length corresponds to the {`size_t`} type declaration.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
<!--ID: 1708426673725-->
END%%
Specifier | Description Specifier | Description
--------- | ----------- --------- | -----------
`d`, `i` | a decimal `signed int` `d`, `i` | a decimal `signed int`
@ -263,7 +321,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`d` and `i`} format specifers work on a {decimal `signed int`}. The {`%d` and `%i`} format specifers work on a {decimal `signed int`}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083040--> <!--ID: 1707852083040-->
@ -271,7 +329,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
{`d` and `i`} are to `signed` as {`u`} is to `unsigned`. {`%d` and `%i`} are to `signed` as {`%u`} is to `unsigned`.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083044--> <!--ID: 1707852083044-->
@ -279,7 +337,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`u`} format specifier outputs a {decimal `unsigned int`}. The {`%u`} format specifier outputs a {decimal `unsigned int`}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083047--> <!--ID: 1707852083047-->
@ -287,7 +345,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What distinguishes format specifiers `d` and `i`? What distinguishes format specifiers `%d` and `%i`?
Back: Nothing. Back: Nothing.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
@ -297,7 +355,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which format specifiers were probably used to yield `printf` output `-12`? Which format specifiers were probably used to yield `printf` output `-12`?
Back: `d` or `i` Back: `%d` or `%i`
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083054--> <!--ID: 1707852083054-->
@ -305,7 +363,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What distinguishes format specifiers `d` and `u`? What distinguishes format specifiers `%d` and `%u`?
Back: The former is for signed integers, the latter unsigned. Back: The former is for signed integers, the latter unsigned.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
@ -314,7 +372,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`x`} format specifier outputs a {lowercase hexadecimal `unsigned int`}. The {`%x`} format specifier outputs a {lowercase hexadecimal `unsigned int`}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083063--> <!--ID: 1707852083063-->
@ -323,7 +381,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which format specifier were probably used to yield `printf` output `7af`? Which format specifier were probably used to yield `printf` output `7af`?
Back: `x` Back: `%x`
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083068--> <!--ID: 1707852083068-->
@ -331,7 +389,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`X`} format specifier outputs an {uppercase hexadecimal `unsigned int`}. The {`%X`} format specifier outputs an {uppercase hexadecimal `unsigned int`}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083074--> <!--ID: 1707852083074-->
@ -340,7 +398,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which format specifier were probably used to yield `printf` output `7AF`? Which format specifier were probably used to yield `printf` output `7AF`?
Back: `X` Back: `%X`
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083080--> <!--ID: 1707852083080-->
@ -348,7 +406,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What distinguishes format specifiers `x` and `X`? What distinguishes format specifiers `%x` and `%X`?
Back: The former outputs lowercase hex digits, the latter uppercase. Back: The former outputs lowercase hex digits, the latter uppercase.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
@ -357,7 +415,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`o`} format specifier outputs an {octal `unsigned int`}. The {`%o`} format specifier outputs an {octal `unsigned int`}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083091--> <!--ID: 1707852083091-->
@ -365,7 +423,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Why doesn't the `o` format specifier have a corresponding `O` specifier? Why doesn't the `%o` format specifier have a corresponding `%O` specifier?
Back: There is no distinction between lower and uppercase octal digits. Back: There is no distinction between lower and uppercase octal digits.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
@ -374,7 +432,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`s`} format specifiers outputs a {`NUL`-terminated string}. The {`%s`} format specifiers outputs a {`NUL`-terminated string}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
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@ -383,7 +441,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which format specifier was probably used to yield `printf` output `abc`? Which format specifier was probably used to yield `printf` output `abc`?
Back: `s` Back: `%s`
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083105--> <!--ID: 1707852083105-->
@ -391,7 +449,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`c`} format specifier outputs a {`char` character}. The {`%c`} format specifier outputs a {`char` character}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083109--> <!--ID: 1707852083109-->
@ -400,7 +458,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which format specifier was probably used to yield `printf` output `a`? Which format specifier was probably used to yield `printf` output `a`?
Back: `c` Back: `%c`
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083113--> <!--ID: 1707852083113-->
@ -408,7 +466,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
The {`p`} format specifier outputs a {`void*` address}. The {`%p`} format specifier outputs a {`void*` address}.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083117--> <!--ID: 1707852083117-->
@ -417,7 +475,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which format specifier was probably used to yield `printf` output `0b80000000`? Which format specifier was probably used to yield `printf` output `0b80000000`?
Back: `p` Back: `%p`
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf
<!--ID: 1707852083121--> <!--ID: 1707852083121-->
@ -425,7 +483,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How is the address outputted by the `p` format specifier written? How is the address outputted by the `%p` format specifier written?
Back: In an implementation-specific way. Back: In an implementation-specific way.
Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962). Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
Tags: printf Tags: printf

74
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@ -238,9 +238,77 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
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END%% END%%
## Bit Strings
A [[binary/index|bit string]] can be used to represent subsets of some finite set. A `1` value usually corresponds to inclusion in a subset, whereas a `0` value corresponds to exclusion. Thus, given set e.g. $A = \{1, 2, 3, 4\}$, $0110_2$ would correspond to subset $\{2, 3\}$.
%%ANKI
Basic
Given $A = \{1, 2, 3, 4\}$, what subset does $1010_2$ correspond to?
Back: $\{1, 3\}$
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: 1708434662140-->
END%%
%%ANKI
Cloze
Representing subsets with bit strings, {$0000_2$} maps to {$\varnothing$} $\subseteq \{1, 2, 3, 4\}$.
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: 1708434662148-->
END%%
%%ANKI
Cloze
Representing subsets with bit strings, {$1111_2$} maps to {$\{1, 2, 3, 4\}$} $\subseteq \{1, 2, 3, 4\}$.
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: 1708434662151-->
END%%
Bit strings also make it clear that the number of subsets with even cardinality must be equal to the number of subsets with odd cardinality. Hence, $$\binom{n}{0} - \binom{n}{1} + \cdots + (-1)^n \binom{n}{n} = 0$$
%%ANKI
Basic
What does $\sum_{k=0}^n (-1)^k \binom{n}{k}$ evaluate to?
Back: $0$
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: 1708434662154-->
END%%
%%ANKI
Basic
What does expression $\sum_{k=0}^n (-1)^k \binom{n}{k} = 0$ indicate about sets?
Back: There is an equal number of even-sized subsets as odd-sized subsets.
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: 1708434662157-->
END%%
%%ANKI
Basic
What isomorphism shows finite sets have an equal number of even-sized subsets as odd-sized subsets?
Back: The one-to-one relationship between subsets and bit strings.
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: 1708434662160-->
END%%
%%ANKI
Basic
Given finite set $A$, how many subsets of even cardinality does $A$ have?
Back: Let $n = |A|$. Then $2^{n-1}$.
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: 1708434662163-->
END%%
%%ANKI
Basic
Given finite set $A$, how many subsets of odd cardinality does $A$ have?
Back: Let $n = |A|$. Then $2^{n-1}$.
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: 1708434662166-->
END%%
## Lattice Paths ## Lattice Paths
A **lattice path** is one of the shorted possible paths connecting two points on a lattice, moving only horizontally and vertically. By representing each horizontal move by "1" and each vertical move by "0", we see every lattice path has a corresponding bit string. A **lattice path** is one of the shorted possible paths connecting two points on a lattice, moving only horizontally and vertically. By representing each horizontal move by `1` and each vertical move by `1`, we see every lattice path has a corresponding [[#Bit Strings|bit string]].
![[lattice-path-example.png]] ![[lattice-path-example.png]]
@ -330,9 +398,9 @@ Back:
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END%% END%%
## Binomials ## Binomial Coefficients
A **binomial** is a polynomial containing two terms. Consider $(x + y)^n$. A **binomial** is a polynomial containing two terms. Consider $(x + y)^n$. We see that term $x^ky^{n-k}$ maps to *some* bit string containing $k$ `1`s and $n - k$ `0`s. This might feel more obvious when expanding to $x \cdot x \cdots x \cdot y \cdots y$. Since multiplication is commutative, the number of matching "bit strings" is the same as $\binom{n}{k}$.
%%ANKI %%ANKI
Basic Basic

96
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@ -0,0 +1,96 @@
---
title: Principle of Inclusion/Exclusion
TARGET DECK: Obsidian::STEM
FILE TAGS: combinatorics set
tags:
- combinatorics
- set
---
## Overview
The **principle of inclusion/exclusion** refers to the oscillating adding and subtracting used to find the cardinality of potentially overlapping sets. Consider sets $A$, $B$, and $C$. Then
$$|A \cup B| = |A| + |B| - |AB|$$
and
$$|A \cup B \cup C| = |A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|$$
Notice the number of terms containing one set, two sets, three sets, etc. match the [[combinations#Binomial Coefficients|binomial coefficients]].
%%ANKI
Basic
Given finite sets $A$ and $B$ and using PIE, what is $|A \cup B|$?
Back: $|AB| = |A| + |B| - |AB|$
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: 1708438356466-->
END%%
%%ANKI
Basic
Given finite sets $A$ and $B$, what combinatorial concept is used to find $|A \cup B|$?
Back: The principle of inclusion/exclusion.
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: 1708438356471-->
END%%
%%ANKI
Basic
Why is the principle of inclusion/exclusion named the way it is?
Back: Because it involves an alternating adding and subtracting of terms.
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: 1708438356474-->
END%%
%%ANKI
Basic
What concept does PIE typically refer to?
Back: The **p**rinciple of **i**nclusion/**e**xclusion.
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: 1708438356477-->
END%%
%%ANKI
Basic
Given finite sets $A$ and $B$, what is $|A \cup B \cup C|$?
Back: $|A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|$
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: 1708438356480-->
END%%
%%ANKI
Basic
Using sigma notation, what identity is used to prove PIE correctly counts members?
Back: $\sum_{k=0}^n (-1)^k \binom{n}{k} = 0$
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: 1708438356483-->
END%%
%%ANKI
Basic
Why might PIE be considered a top-down approach to counting?
Back: It starts by counting every member of each union operand.
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: 1708438356486-->
END%%
%%ANKI
Basic
What is the bottom-up approach contrasting PIE?
Back: Apply the additive property to all disjoint sets the union operands can make.
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: 1708438356490-->
END%%
%%ANKI
Basic
Given finite sets $A$ and $B$ and using a bottom-up approach (i.e. *not* PIE), what is $|A \cup B|$?
Back: $|A \cup B| = |AB| + |A - AB| + |B - AB|$
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: 1708438356493-->
END%%
## References
* 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).