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b/Spring-2023/CS-2233/Assignment-1/Solution.typ @@ -0,0 +1,642 @@ +#let m(math) = align(center)[$#math$] +#let pgbreakmsg = align(center, text(blue, weight: "black", size: 1.5em)[See Next Page\ ↓]) + +#let solve(work, solution) = align( + center, +)[ + #let solution = align(center, block( + inset: 5pt, + stroke: blue + .3pt, + fill: rgb(0, 149, 255, 15%), + radius: 4pt, + )[#align(left)[#solution]]) + + #if work == [] [ + #solution + ] else [ + #block(inset: 6pt, radius: 4pt, stroke: luma(50%) + .5pt, fill: luma(90%))[ + #align(left, text(font: "Liberation Sans", size: .85em, work)) + #solution + ] + ] +] + +#let problem-header(number, points) = [== Problem #number. #text(weight: "regular")[[#points + points]]] + +#let problem(number, points, body) = [ + == Problem #number. #text(weight: "regular")[[#points points]] + #body +] + +#set page(margin: (x: .4in, y: .4in)) +#set table(align: center) + +_*Price Hiller*_ +#v(-.8em) +_*zfp106*_ +#v(-.8em) +Homework Assignment 1 +#v(-.8em) +CS 2233 +#v(-.8em) +Section 001 + +#align( + center, + block( + inset: 6pt, + radius: 4pt, + stroke: luma(50%) + .5pt, + fill: luma(90%), + )[If you are interested in viewing the source code of this document, you can do so by clicking + #text( + blue, + link( + "https://gitlab.orion-technologies.io/philler/college/-/blob/Development/Spring-2023/CS-2233/Assignment-1/Solution.typ?ref_type=heads", + "here", + ), + ). This document was written in Typst and a bit of infinite _withering_ pain in Neovim, a Vim + derivative. Here's to hoping everything below is correct.], +) += Problems + +#problem( + 1, + 10, +)[ + - Complete all participation activities in zyBook sections: 1.1, 1.2, 1.3, 1.4, + 1.5. + #solve[][Done] +] + +#problem( + 2, + 15, +)[ + Let $p$ denote "You passed CS 2233". + + Let $q$ denote "You passed CS 3333". + + Let $r$ denote "You can register for CS 3343". + + Let $s$ denote "You understand propsitional logic". + + Use $p$, $q$, $r$, and $s$, to create propositions representing the following + statements. + + a. [5 points] You did not pass CS 2233, but you understand propositional logic. + + #solve[ + This can be alternatively expressed as + #align( + center, + )["You did not pass CS 2233 and you understand propositional logic"] + As the "but" in the statement is _not_ an exclusion, it is a conjunction in + typical english language. + ][$¬p ∧ s$] + + b. [5 points] You cannot register for CS 3343 only if you have not passed both + CS 2233 and CS 3333 + + #solve[ + This can be alternatively expressed as + #align( + center, + )["If you can register for CS 3343 then that implies you have passed CS 2233 and + CS 3333"] + Which can also be rewritten as + #align( + center, + )["If you have passed CS 2233 and CS 3333 then you can register for CS 3343"] + + In the original statement, we're negating both sides, thus $¬r$ and $¬p ∧ ¬q$ which + can be written as: + ][$¬r → ¬(p ∧ q)$] + + #pgbreakmsg + #pagebreak() + c. [5 points] If you can register for CS 3343, then you have passed CS 2233 and + you understand propositional logic if you passed CS 2233\ + + #solve[ + So this can be more easily understood by "solving" the latter half of the + statement first. + + Expressing "You understand propositional logic if you passed CS 2233" logically + would be: + #m[s → q] + + Now slightly rewriting the statement with our logic embedded, "If you can + register for CS 3343, then you have passed CS 2233 and $(s → q)$". + + Then further refining the statement: "If you can register for CS 3343 then $p ∧ (s → q)$. + + And with a final refinement: $r → p ∧ (s → q)$ + + Thus the final expression logically would be: + ][$r → p ∧ (s → p)$] +] +#problem( + 3, + 40, +)[ + Show that $(¬q ∧ (p ∨ p)) → ¬q$ is a tautology, i.e. $(¬q ∧ (p ∨ p)) → ¬q ≡ T$ + + a. [10 points] By creating a truth table + + // | q | p | ¬q | (p ∨ p) | (¬q ∧ (p ∨ p)) | (¬q ∧ (p ∨ p)) → ¬q | + // |---|---|----|---------|----------------|---------------------| + // | T | T | F | T | F | T | + // | T | F | F | F | F | T | + // | F | T | T | T | T | T | + // | F | F | T | F | F | T | + #solve[ + Notice that the furthest right column only has true values, thus the above + proposition is a + _tautology_ and is always _True_ + ][ + #table( + columns: 6, + [q], + [p], + [¬q], + [(p ∨ p)], + [(¬q ∧ (p ∨ p))], + [(¬q ∧ (p ∨ p)) → ¬q], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + ) + ] + b. [10 points] By creating a sequence of logical equivalences and annotating + each step + + #solve[][ + #grid(columns: 2, row-gutter: .5em, column-gutter: 6em)[ + 1. $¬q ∧ (p ∨ p) → ¬q$ + ][Starting Proposition][ + 2. $(¬q ∧ p) → ¬q ∵ (z ∨ z) ≡ p$ + ][Idempotent Law][ + 3. $¬(¬q ∧ p) ∨ ¬q ∵ y → z ≡ ¬y ∨ q$ + ][Conditional Identitify][ + 4. $(¬¬q ∨ ¬p) ∨ q ∵ ¬(z ∧ y) ≡ (¬z ∨ ¬q) $ + ][De Morgan's laws][ + 5. $(q ∨ ¬p) ∨ ¬q ∵ ¬¬z ≡ z$ + ][Double Negation laws][ + 6. $(q ∨ ¬q) ∨ ¬p ∵ (z ∨ y) ∨ x ≡ z ∨ (y ∨ x)$ + ][Associative Laws][ + 7. $T ∨ ¬p ∵ z ∨ ¬z ≡ T ∴ (q ∨ ¬q) ∨ ¬p ≡ T ∨ ¬p$ + ][Complement Laws][ + 8. $T ∵ T ∨ z ≡ T$ + ][Domination Laws] + ] + #pgbreakmsg + #pagebreak(weak: true) + + Show that $¬q → (p ∧ r) ≡ (¬q → r) ∧ (q ∨ p)$ + + c. [10 points] By creating a truth table + + // * ¬q → (p ∧ r) + // + // | $q$ | $p$ | $r$ | $¬q$ | $(p ∧ r)$ | $¬q → (p ∧ r)$ | + // |-----|-----|-----|------|-----------|----------------| + // | T | T | T | F | T | T | + // | T | T | F | F | F | T | + // | T | F | T | F | F | T | + // | T | F | F | F | F | T | + // | F | T | T | T | T | T | + // | F | T | F | T | F | F | + // | F | F | T | T | F | F | + // | F | F | F | T | F | F | + // + // + // * (¬q → r) ∧ (q ∨ p) + // + // | $q$ | $p$ | $r$ | $¬q$ | $¬q → r$ | $q ∨ p$ | $(¬q → r) ∧ (q ∨ p)$ | + // |-----|-----|-----|------|----------|---------|----------------------| + // | T | T | T | F | T | T | T | + // | T | T | F | F | T | T | T | + // | T | F | T | F | T | T | T | + // | T | F | F | F | T | T | T | + // | F | T | T | T | T | T | T | + // | F | T | F | T | F | T | F | + // | F | F | T | T | T | F | F | + // | F | F | F | T | T | F | F | + #solve[ + Notice that the furthest right column only has true values, thus the above + statement is a + _tautology_ and is always _True_ + ][ + #table(columns: 2, stroke: none, [Truth table for *$¬q → (p ∧ r)$* + #table( + columns: 6, + [$q$], + [$p$], + [$r$], + [$¬q$], + [$(p ∧ r)$], + [$¬q → (p ∧ r)$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + )], [Truth table for *$(¬q → r) ∧ (q ∨ p)$* + #table( + columns: 7, + [$q$], + [$p$], + [$r$], + [$¬q$], + [$¬q → r$], + [$q ∨ p$], + [$(¬q → r) ∧ (q ∨ p)$ ], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + )]) + ] + + d. [10 points] By creating a sequence of logical equivalences and annotating + each step + #solve[Notice by step three, both logical sequences are equivalent.][ + #table(columns: 2, [ + *$¬q → (p ∧ r)$* + #grid(columns: 2, row-gutter: .5em, column-gutter: 1em)[ + 1. $¬¬q ∨ (p ∧ r)$ + ][Conditional Identities][ + 2. $q ∨ (p ∧ r)$ + ][Double Negation Law][ + 3. $(q ∨ p) ∧ (q ∨ r)$ + ][Distributive Laws] + ], [ + #align(left)[*$(¬q → r) ∧ (q ∨ p)$*] + #grid(columns: 2, row-gutter: .5em, column-gutter: 1em)[ + 1. $(¬¬q ∨ r) ∧ (q ∨ p)$ + ][Conditional Identities][ + 2. $(q ∨ r) ∧ (q ∨ p)$ + ][Double Negation Law][ + 3. $(q ∨ p) ∧ (q ∨ r)$ + ][Commutative Laws] + ]) + Thus $¬q → (p ∧ r) ≡ (¬q → r) ∧ (q ∨ p)$ + ] +] + +#pgbreakmsg +#pagebreak() +#problem( + 4, + 20, +)[ + a. [10 points] Show that the $∨$ operator is associative by creating a truth + table showing that $p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r$ + + #solve[Notice that the furthest right columns of both tables are equivalent, therefore + the $∨$ operator is associative][ + #table(columns: 2, stroke: none, [Truth table for *$p ∨ (q ∨ r)$* + #table( + columns: 5, + [$q$], + [$p$], + [$r$], + [$(q ∨ r)$], + [$p ∨ (q ∨ r)$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$F$], + )], [Truth table for *$(p ∨ q) ∨ r$* + #table( + columns: 5, + [$q$], + [$p$], + [$r$], + [$(p ∨ q)$], + [$(p ∨ q) ∨ r$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$T$], + [$T$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$F$], + )]) + ] + + b. [10 points] The NOR operator $↓$ is the negation of a disjunction: $p ↓ q ≡ ¬(p ∨ q)$. + Its truth table is: + + #table( + columns: 3, + [$p$], + [$q$], + [$p ↓ q$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + ) + + Show that The NOR operator is not associative by creating a truth table showing + that it is not the case that *$p ↓ (q ↓ r) ≡ (p ↓ q) ↓ r$*. In other words, + create a truth table showing that *$(p ↓ (q ↓ r)) ↔ ((p ↓ q) ↓ r)$* is not a + tautology. + + #solve[Notice that the two tables' outputs are different in the furthest righthand + column. If the NOR were associative, the furthest right columns of both tables + would be identical. Since this is not the case, the NOR operator isn't + associative.][#table(columns: 2, stroke: none, [Truth table for *$p ↓ (q ↓ r)$* + #table( + columns: 5, + [$q$], + [$p$], + [$r$], + [$(q ↓ r)$], + [$p ↓ (q ↓ r)$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + )], [Truth table for *$(p ↓ q) ↓ r$* + #table( + columns: 5, + [$q$], + [$p$], + [$r$], + [$(p ↓ q)$], + [$(p ↓ q) ↓ r$], + [$T$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + [$F$], + [$F$], + [$T$], + [$T$], + [$F$], + [$F$], + [$F$], + [$F$], + [$T$], + [$F$], + )])] +]