Encoding and inclusion/exclusion principle.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

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-    "binary/integer-encoding.md": "a2c8c83a20f1124fd5af0f3c23894284",
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notes/_journal/2024-02-20.md

@@ -7,8 +7,8 @@ title: "2024-02-20"
 - [ ] Sheet Music (10 min.)
 - [x] 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)
+- [x] Log Work Hours (Max 3 hours)
 
 * Added `printf` `length` field notes.
 * 101weiqi (serial numbers)
@@ -19,3 +19,4 @@ title: "2024-02-20"
 	* Q-275961
 	* Q-324650
 	* Q-83832
+* Notes on integer encodings and how unsigned encoding relates to two's-complement.

notes/binary/integer-encoding.md

@@ -8,7 +8,15 @@ tags:
 
 ## Overview
 
-Integers are typically encoded using either **unsigned encoding** or **two's complement**.
+Integers are typically encoded using either **unsigned encoding** or **two's complement**. The following table highlights how the min and max of these encodings behave:
+
+Value    | $w = 8$ | $w = 16$ | $w = 32$
+-------- | ------- | -------- | ------------
+$UMin_w$ | `0x00`  | `0x0000` | `0x00000000`
+$UMax_w$ | `0xFF`  | `0xFFFF` | `0xFFFFFFFF`
+$TMin_w$ | `0x80`  | `0x8000` | `0x80000000`
+$TMax_w$ | `0x7F`  | `0x7FFF` | `0x7FFFFFFF`
+
 
 %%ANKI
 Basic
@@ -68,6 +76,38 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708177246114-->
 END%%
 
+%%ANKI
+Basic
+Which of unsigned encoding or two's-complement exhibit asymmetry in their range?
+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: 1708453398379-->
+END%%
+
+%%ANKI
+Basic
+What integral values share the same binary representation in unsigned encoding and two's-complement?
+Back: Nonnegative values $\leq |TMax|$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708454709515-->
+END%%
+
+%%ANKI
+Basic
+According to the C standard, how are `unsigned` integral types encoded?
+Back: Using unsigned encoding.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708455064691-->
+END%%
+
+%%ANKI
+Basic
+According to the C standard, how are `signed` integral types encoded?
+Back: The C standard leaves this unspecified.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708455064696-->
+END%%
+
 ### Unsigned Encoding
 
 Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$$
@@ -170,10 +210,34 @@ END%%
 
 %%ANKI
 Basic
-Why is "$B2U$" insufficient for use?
+What is the hexadecimal representation of $UMin_4$?
-Back: We need to understand how many bits conversion is with respect to.
+Back: `0x0000`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708179147804-->
+<!--ID: 1708453398386-->
+END%%
+
+%%ANKI
+Basic
+How is the smallest unsigned integer formatted in hexadecimal?
+Back: As all `0`s.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398392-->
+END%%
+
+%%ANKI
+Basic
+What is the hexadecimal representation of $UMax_4$?
+Back: `0xFFFF`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398397-->
+END%%
+
+%%ANKI
+Basic
+How is the largest unsigned integer formatted in hexadecimal?
+Back: As all `F`s.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398403-->
 END%%
 
 ### Two's-Complement
@@ -293,10 +357,98 @@ END%%
 
 %%ANKI
 Basic
-Why is "$B2T$" insufficient for use?
+What is the hexadecimal representation of $TMin_4$?
-Back: We need to understand how many bits conversion is with respect to.
+Back: `0x8000`
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708179649935-->
+<!--ID: 1708453398408-->
+END%%
+
+%%ANKI
+Basic
+How is the smallest two's-complement integer formatted in hexadecimal?
+Back: With a leading `8` followed by `0`s.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398413-->
+END%%
+
+%%ANKI
+Basic
+What is the hexadecimal representation of $TMax_4$?
+Back: `0x7FFF`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398418-->
+END%%
+
+%%ANKI
+Basic
+How is the largest two's-complement integer formatted in hexadecimal?
+Back: With a leading `7` followed by `F`s.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398425-->
+END%%
+
+%%ANKI
+Basic
+How is equality $|TMin| = |TMax|$ modified so that both sides actually balance?
+Back: $|TMin| = |TMax| + 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398430-->
+END%%
+
+%%ANKI
+Basic
+Which of negative and positive numbers can two's-complement encoding express more of?
+Back: Negative.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398435-->
+END%%
+
+%%ANKI
+Basic
+Why is two's-complement encoding's range asymmetric?
+Back: Leading `1`s correspond to negatives but leading `0`s corerspond to nonnegative numbers (which include $0$.)
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398440-->
+END%%
+
+%%ANKI
+Basic
+How does $UMax$ relate to $TMax$?
+Back: $|UMax| = 2|TMax| + 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398445-->
+END%%
+
+%%ANKI
+Basic
+What are the binary encodings of $UMax_4$ and $TMax_4$?
+Back: $1111_2$ and $0111_2$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398449-->
+END%%
+
+%%ANKI
+Basic
+Reinterpret $TMax$ in unsigned encoding. What arithmetic operations yield $UMax$?
+Back: Multiply by two and add one.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398454-->
+END%%
+
+%%ANKI
+Basic
+Reinterpret $TMax$ in unsigned encoding. What bitwise operations yield $UMax$?
+Back: One-bit left shift and add one.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398459-->
+END%%
+
+%%ANKI
+Basic
+Reinterpret $UMax$ in two's-complement. What decimal value do you have?
+Back: $-1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398469-->
 END%%
 
 ## References

notes/c/strings.md

@@ -490,6 +490,68 @@ Tags: printf
 <!--ID: 1707852083126-->
 END%%
 
+%%ANKI
+Basic
+Given `int64_t x`, why is `printf("%d", x)` a problem?
+Back: `%d` matches an `int` which is not necessarily 64-bits.
+Tags: printf
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708454462772-->
+END%%
+
+%%ANKI
+Basic
+What must you use when invoking `printf` with a fixed-width integer type? 
+Back: `printf`-specific macros.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: printf
+<!--ID: 1708454462777-->
+END%%
+
+%%ANKI
+Basic
+What is `PRId32` an example macro for?
+Back: A macro that expands to the correct specifier for a 32-bit signed integral type.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: printf
+<!--ID: 1708454462780-->
+END%%
+
+%%ANKI
+Cloze
+{`PRId32`} is to signed whereas {`PRIu32`} 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).
+Tags: printf
+<!--ID: 1708454462784-->
+END%%
+
+%%ANKI
+Basic
+Which C header specifies `printf` macros for fixed-width integral types?
+Back: `<inttypes.h>`
+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
+<!--ID: 1708454462788-->
+END%%
+
+%%ANKI
+Basic
+Given `int32_t x`, how might we invoke `printf` on it?
+Back: `printf("%" PRId32, x)`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: printf
+<!--ID: 1708454584564-->
+END%%
+
+%%ANKI
+Basic
+What prefix do `printf` macros from `<inttypes.h>` share?
+Back: `PRI`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: printf
+<!--ID: 1708454584568-->
+END%%
+
 ## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/combinatorics/combinations.md

@@ -160,6 +160,46 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708368078746-->
 END%%
 
+%%ANKI
+Basic
+How many *increasing* injective functions exist between $\{1, 2, 3\}$ and $\{a, b, c, d, e\}$?
+Back: $\binom{5}{3}$
+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: 1708446818783-->
+END%%
+
+%%ANKI
+Basic
+How many *decreasing* injective functions exist between $\{1, 2\}$ and $\{a, b, c, d\}$?
+Back: $\binom{4}{2}$
+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: 1708446818786-->
+END%%
+
+%%ANKI
+Basic
+Given finite sets $A$ and $B$, what is the number of *increasing* injective functions between $A$ and $B$?
+Back: Given $k = |A|$ and $n = |B|$, $\binom{n}{k}$.
+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: 1708446818788-->
+END%%
+
+%%ANKI
+Basic
+What combinatorial concept explains the number of *increasing* injective functions between two finite sets?
+Back: Combinations.
+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: 1708446818789-->
+END%%
+
+%%ANKI
+Basic
+Given $k = |A|$ and $n = |B|$, *why* is the number of increasing injective functions between $A$ and $B$ equal to $\binom{n}{k}$?
+Back: We are "grouping" all functions by a shared permutation (i.e. the increasing function).
+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: 1708446818791-->
+END%%
+
 ## Pascal's Triangle
 
 A visual representation of the binomial coefficient's is in the form of Pascal's Triangle:
@@ -183,6 +223,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708384441360-->
 END%%
 
+%%ANKI
+Basic
+*Why* is it that $\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$?
+Back: For each member, we either include in a subset or we don't.
+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: 1708446818792-->
+END%%
+
 %%ANKI
 Basic
 What name is given to the following structure?
@@ -314,6 +362,14 @@ A **lattice path** is one of the shorted possible paths connecting two points on
 
 In this example, the total number of lattice paths from point $(0, 0)$ to $(3, 2)$ is therefore $\binom{5}{2} = \binom{5}{3}$.
 
+%%ANKI
+Basic
+How many lattice paths are there from $(0, 0)$ to $(n, n)$?
+Back: $\binom{2n}{n}$
+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: 1708451749788-->
+END%%
+
 %%ANKI
 Basic
 What is the integer lattice?
@@ -450,6 +506,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708384441462-->
 END%%
 
+%%ANKI
+Basic
+How is $7^n$ written as a sum of powers of $6$?
+Back: $7^n = (1 + 6)^n$. Apply binomial expansion on the RHS.
+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: 1708451749791-->
+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).

notes/combinatorics/inclusion-exclusion.md

@@ -61,7 +61,7 @@ END%%
 
 %%ANKI
 Basic
-Using sigma notation, what identity is used to prove PIE correctly counts members?
+Using sigma notation, what binomial 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-->

notes/combinatorics/permutations.md

@@ -61,6 +61,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708366788573-->
 END%%
 
+%%ANKI
+Basic
+How is $n!$ written recursively?
+Back: As $n(n - 1)!$.
+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: 1708451749781-->
+END%%
+
 %%ANKI
 Basic
 How is permutation expressed recursively?