Little o- and omega-notation.

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

@@ -135,7 +135,7 @@
     "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
     "algorithms/loop-invariant.md": "29f9f9090a3109890d333a78acc18b50",
     "algorithms/running-time.md": "5efc0791097d2c996f931c9046c95f65",
-    "algorithms/order-growth.md": "f4bab009ba9a6af34b9245fe595b0a2a",
+    "algorithms/order-growth.md": "cf80f0fddaedbfb1959d05adbfe6cbee",
     "_journal/2024-02-08.md": "19092bdfe378f31e2774f20d6afbfbac",
     "algorithms/sorting/selection-sort.md": "fcd0dc2ebaabd0a4db1baf7e7ef9f7a9",
     "algorithms/index 1.md": "6fada1f3d5d3af64687719eb465a5b97",
@@ -222,7 +222,7 @@
     "filesystems/cas.md": "34906013a2a60fe5ee0e31809b4838aa",
     "git/objects.md": "b95228a78744d3f9fe173e575aa0445a",
     "git/index.md": "83d2d95fc549d9e8436946c7bd058d15",
-    "encoding/integer.md": "3107074a5070670f2f8fab17abc05fbb",
+    "encoding/integer.md": "84a8343b1cf3cd1552c9fce3ef3e52b8",
     "_journal/2024-02-29.md": "f610f3caed659c1de3eed5f226cab508",
     "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82",
     "_journal/2024-03-01.md": "a532486279190b0c12954966cbf8c3fe",
@@ -232,7 +232,9 @@
     "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1",
     "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322",
     "algebra/sequences/triangular-numbers.md": "18925b0ecae151c3e6d38bc018c632c4",
-    "algebra/sequences/square-numbers.md": "886fb22fb8dbfffdd2cd233558ea3424"
+    "algebra/sequences/square-numbers.md": "886fb22fb8dbfffdd2cd233558ea3424",
+    "_journal/2024-03-03.md": "c4977a3778ed227b768c3f9ad5512670",
+    "_journal/2024-03/2024-03-02.md": "8136792b0ee6e08232e4f60c88d461d2"
   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-03-03.md

@@ -0,0 +1,14 @@
+---
+title: "2024-03-03"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [ ] Interview Prep (1 Practice Problem)
+- [ ] Log Work Hours (Max 3 hours)
+
+* Began working through unsigned and two's-complement multiplication.
+* Added notes on $o$- and $\omega$-notation.

notes/algorithms/order-growth.md

@@ -169,6 +169,51 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1708974221871-->
 END%%
 
+When encountering equations with asymptotic notation on both sides of the equality, we interpret the equation using the following rule:
+
+> No matter how the anonymous functions are chosen on the left of the equal sign, there is a way to choose the anonymous functions on the right of the equal sign to make the equation valid.
+
+For example, $2n^2 + \Theta(n) = \Theta(n^2)$ states that for all $f(n) \in \Theta(n)$, there exists some $g(n) \in \Theta(n^2)$ such that $2n^2 + f(n) = g(n)$.
+
+%%ANKI
+Basic
+Asymptotic notation on the RHS of an equation is a stand in for what?
+Back: *Some* function in the set that satisfies the equation.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002305-->
+END%%
+
+%%ANKI
+Basic
+Asymptotic notation on the LHS of an equation is a stand in for what?
+Back: *Any* function in the set.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002309-->
+END%%
+
+%%ANKI
+Cloze
+In equations containing asymptotic notation, {1:LHS} is to {1:$\forall$} whereas {2:RHS} is to {2:$\exists$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002313-->
+END%%
+
+%%ANKI
+Basic
+How is $2n^2 + \Theta(n) = \Theta(n^2)$ written in propositional logic?
+Back: $\forall f(n) \in \Theta(n), \exists g(n) \in \Theta(n^2), 2n^2 + f(n) = g(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002316-->
+END%%
+
+%%ANKI
+Basic
+*Why* is $\sum_{i=1}^n O(i) \neq O(1) + O(2) + \cdots + O(n)$?
+Back: The number of anonymous functions is equal to the number of times the asymptotic notation appears.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002319-->
+END%%
+
 ## $\Theta$-notation
 
 ![[theta-notation.png]]
@@ -437,7 +482,7 @@ END%%
 
 %%ANKI
 Basic
-Which notation corresponds to asymptotic upper bounds?
+Which notation corresponds to (potentially tight) asymptotic upper bounds?
 Back: $O$-notation.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
 <!--ID: 1709053894088-->
@@ -509,6 +554,112 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1709053894105-->
 END%%
 
+## $o$-notation
+
+$o$-notation refers to an upper bound that is not asymptotically tight. It is defined as set $$o(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq f(n) < cg(n) \}$$
+
+%%ANKI
+Basic
+What kind of mathematical object is $o(g(n))$?
+Back: A set.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002323-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers found in $o(g(n))$, what values do $c$ and $n_0$ take on?
+Back: Positive constants.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002325-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the existentially quantified identifers in $o(g(n))$'s definition?
+Back: $n_0$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002328-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the universally quantified identifers in $o(g(n))$'s definition?
+Back: $c$ and $n$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002331-->
+END%%
+
+%%ANKI
+Cloze
+Using typical identifiers, $f(n) = o(g(n))$ satisfies {$0$} $\leq$ {$f(n)$} $<$ {$cg(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002334-->
+END%%
+
+%%ANKI
+Basic
+How does $o$-notation compare to $O$-notation?
+Back: The former denotes an upper bound that is not asymptotically tight.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002337-->
+END%%
+
+%%ANKI
+Basic
+How is $o(g(n))$ pronounced?
+Back: As "little-oh of $g$ of $n$".
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002340-->
+END%%
+
+%%ANKI
+Basic
+How can $f(n) = o(g(n))$ be expressed as a limit?
+Back: $$\lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002344-->
+END%%
+
+%%ANKI
+Basic
+Which notation corresponds to asymptotic upper bounds that are not tight?
+Back: $o$-notation.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002347-->
+END%%
+
+%%ANKI
+Basic
+Every member of $o(g(n))$ is expected to be asymptotically what?
+Back: Nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002350-->
+END%%
+
+%%ANKI
+Basic
+How is $o(g(n))$ defined?
+Back: $\{ \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq f(n) < cg(n) \}$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002353-->
+END%%
+
+%%ANKI
+Cloze
+In $O(g(n))$, bound {1:$0 \leq f(n) \leq cg(n)$} holds for {1:some $c > 0$}. In $o(g(n))$, {2:$0 \leq f(n) < cg(n)$} holds for {2:all $c > 0$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002359-->
+END%%
+
+%%ANKI
+Basic
+Is $O$-notation considered stronger or weaker than $o$-notation?
+Back: Weaker.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002364-->
+END%%
+
 ## $\Omega$-notation
 
 ![[big-omega-notation.png]]
@@ -595,7 +746,7 @@ END%%
 
 %%ANKI
 Basic
-Which notation corresponds to asymptotic lower bounds?
+Which notation corresponds to (potentially tight) asymptotic lower bounds?
 Back: $\Omega$-notation.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
 <!--ID: 1709055157393-->
@@ -665,6 +816,112 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1709055157406-->
 END%%
 
+## $\omega$-notation
+
+$\omega$-notation refers to a lower bound that is not asymptotically tight. It is defined as set $$\omega(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) < f(n) \}$$
+
+%%ANKI
+Basic
+What kind of mathematical object is $\omega(g(n))$?
+Back: A set.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002369-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers found in $\omega(g(n))$, what values do $c$ and $n_0$ take on?
+Back: Positive constants.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002374-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the existentially quantified identifers in $\omega(g(n))$'s definition?
+Back: $n_0$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002380-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the universally quantified identifers in $\omega(g(n))$'s definition?
+Back: $c$ and $n$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002385-->
+END%%
+
+%%ANKI
+Cloze
+Using typical identifiers, $f(n) = \omega(g(n))$ satisfies {$0$} $\leq$ {$cg(n)$} $<$ {$f(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002390-->
+END%%
+
+%%ANKI
+Basic
+How does $\omega$-notation compare to $\Omega$-notation?
+Back: The former denotes a lower bound that is not asymptotically tight.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002395-->
+END%%
+
+%%ANKI
+Basic
+How is $\omega(g(n))$ pronounced?
+Back: As "little-omega of $g$ of $n$".
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002398-->
+END%%
+
+%%ANKI
+Basic
+How can $f(n) = \omega(g(n))$ be expressed as a limit?
+Back: $$\lim_{n \to \infty} \frac{f(n)}{g(n)} = \infty$$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002402-->
+END%%
+
+%%ANKI
+Basic
+Which notation corresponds to asymptotic lower bounds that are not tight?
+Back: $\omega$-notation.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002406-->
+END%%
+
+%%ANKI
+Basic
+Every member of $\omega(g(n))$ is expected to be asymptotically what?
+Back: Nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002410-->
+END%%
+
+%%ANKI
+Basic
+How is $\omega(g(n))$ defined?
+Back: $\{ \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) < f(n) \}$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002414-->
+END%%
+
+%%ANKI
+Cloze
+In $\Omega(g(n))$, bound {1:$0 \leq cg(n) \leq f(n)$} holds for {1:some $c > 0$}. In $\omega(g(n))$, {2:$0 \leq cg(n) < f(n)$} holds for {2:all $c > 0$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002420-->
+END%%
+
+%%ANKI
+Basic
+Is $\omega$-notation considered stronger or weaker than $\Omega$-notation?
+Back: Stronger.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709519002425-->
+END%%
+
 ## References
 
 * Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).

notes/encoding/integer.md

@@ -720,6 +720,13 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708615249914-->
 END%%
 
+%%ANKI
+Cloze
+For all $x$, $T2U_w(x)=$ {$x + x_{w-1}2^w$} where $x_{w-1}$ is the most significant bit of $x$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205954-->
+END%%
+
 %%ANKI
 Basic
 How is $T2U_w$ written as a function composition?
@@ -1083,7 +1090,7 @@ END%%
 
 %%ANKI
 Basic
-Why can we compare $s$ to $x$ or $y$ when detecting overflow of $s \coloneqq x +_w^u y$?
+Why can we choose to compare $s$ to either $x$ or $y$ when detecting overflow of $s \coloneqq x +_w^u y$?
 Back: Because unsigned addition is commutative.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708799678776-->
@@ -1121,6 +1128,22 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1709042784788-->
 END%%
 
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the largest possible value of $x +_w^u y$?
+Back: $w + 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205961-->
+END%%
+
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the smallest possible value of $x +_w^u y$?
+Back: $w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205964-->
+END%%
+
 Two's-complement addition, denoted $+_w^t$ operates similarly:
 
 $$x +_w^u y = \begin{cases}
@@ -1141,7 +1164,7 @@ END%%
 
 %%ANKI
 Basic
-Why is two's-complement addition overflow UB?
+Why is signed addition overflow UB?
 Back: Because the C standard does not mandate any particular signed integer encoding.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: c17
@@ -1209,6 +1232,14 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708964376254-->
 END%%
 
+%%ANKI
+Basic
+How is the following expressed more simply (i.e. using more standard algebra)? $$x +_w^t y = U2T_w(T2U_w(x) +_w^u T2U_w(y))$$
+Back: $x +_w^t y = U2T_w((x + y) \bmod 2^w)$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205967-->
+END%%
+
 %%ANKI
 *Why* are we able to characterize $+_w^t$ in terms of $+_w^u$?
 Back: Because two's-complement addition has the same bit-level representation as unsigned addition.
@@ -1320,6 +1351,135 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1709044103781-->
 END%%
 
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the largest possible value of $x +_w^t y$?
+Back: $w + 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205970-->
+END%%
+
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the smallest possible value of $x +_w^t y$?
+Back: $w + 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205974-->
+END%%
+
+### Multiplication
+
+Unsigned multiplication, denoted with the $*_w^u$ operator, is defined as follows: $$x *_w^u y = (x \cdot y) \bmod 2^w$$
+
+%%ANKI
+Basic
+What does $*_w^u$ denote?
+Back: Unsigned multiplication of $w$-bit integral types.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205977-->
+END%%
+
+%%ANKI
+Basic
+What is the result of $x *_w^u y$?
+Back: $(x \cdot y) \bmod 2^w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205981-->
+END%%
+
+%%ANKI
+Basic
+*Why* does $x *_w^u y = (x \cdot y) \bmod 2^w$ (at least in C)?
+Back: Because unsigned multiplication is *defined* to be the result truncated to $w$ bits.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1709492205984-->
+END%%
+
+%%ANKI
+Basic
+How do $+_w^u$ and $*_w^u$ behave similarly?
+Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^u\; y = (x \;\square\; y) \bmod 2^w$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1709492205988-->
+END%%
+
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the largest possible value of $x *_w^u y$?
+Back: $2w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205991-->
+END%%
+
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the smallest possible value of $x *_w^u y$?
+Back: $w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205995-->
+END%%
+
+Similarly, two's-complement multiplication is defined as follows: $$x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$$
+
+%%ANKI
+Basic
+What does $*_w^t$ denote?
+Back: Two's-complement multiplication of $w$-bit integral types.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492205998-->
+END%%
+
+%%ANKI
+Basic
+What is the result of $x *_w^t y$?
+Back: $U2T_w((x \cdot y) \bmod 2^w)$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492206002-->
+END%%
+
+%%ANKI
+Basic
+How do $+_w^t$ and $*_w^t$ behave similarly?
+Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^t\; y = U2T_w((x \;\square\; y) \bmod 2^w)$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1709492206006-->
+END%%
+
+%%ANKI
+Basic
+How can we write $x *_w^t y$ in terms of unsigned multiplication?
+Back: $x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492206012-->
+END%%
+
+%%ANKI
+Basic
+How is the following expressed more simply (i.e. using more standard algebra)? $$x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$$
+Back: $x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492206017-->
+END%%
+
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the largest possible value of $x *_w^t y$?
+Back: $2w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492206024-->
+END%%
+
+%%ANKI
+Basic
+Ignoring overflow, what is the width of the smallest possible value of $x *_w^t y$?
+Back: $2w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709492206031-->
+END%%
+
 ### Shifting
 
 Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways: