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Assignment 2

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# File week-01/assignment-02.markdown

`+Assignment 2`
`+============`
`+`
`+Problems`
`+--------`
`+`
`+### Problem 1`
`+`
`+1. `(π>0)∧(π<10)`: `0 < π < 10``
`+2. `(p≥7)∧(p<12)`: `7 ≤ p < 12``
`+3. `(x>5)∧(x<7)`: `5 < x < 7``
`+4. `(x<4)∧(x<6)`: `x < 4``
`+5. `(y<4)∧(y^2<9)`: `-3 < y < 3``
`+6. `(x≥0)∧(x≤0)`: `x = 0``
`+`
`+### Problem 2`
`+`
`+1. π is greater than 0 and less than 10.`
`+2. p is greater than or equal to 7, but less than 12.`
`+3. x is greater than 5 but less than 7.`
`+4. x is less than 4.  The `(x<6)` clause is unneeded because anything less than`
`+   4 is also less than 6.`
`+5. y is greater than -3 but less than 3.  This requirement comes from the `(y^2`
`+   < 9)` clause.  Anything larger than 3^2 is larger than 9.  Anything smaller`
`+   than -3^2 is also larger than 9.  The `(y < 4)` clause is irrelevant, because`
`+   we already know y must be less than 3.`
`+6. x equals zero.  This is the only possible number that can satisfy both`
`+   conditions at once.`
`+`
`+### Problem 3`
`+`
`+If φ1 through φn are related in some way, start by proving the base case and`
`+then prove by induction that φ necessarily implies φn+1.`
`+`
`+If they're not related (I'm not sure exactly what this question is asking) then`
`+you have to just prove them all.  I'd start with the one that looked most likely`
`+to be false since if it *does* turn out to be false I can do a lot less work.`
`+`
`+### Problem 4`
`+`
`+Similar to problem 4: start with the piece most likely to be false, which will`
`+let you short-circuit quickly.`
`+`
`+### Problem 5`
`+`
`+1. `(π>3)∨(π>10)`: `(π > 3) ∨ (π > 10)``
`+2. `(x<0)∨(x>0)`: `x ≠ 0``
`+3. `(x=0)∨(x>0)`: `x ≥ 0``
`+4. `(x>0)∨(x≥0)`: `x ≥ 0``
`+5. `(x>3)∨(x^2>9)`: `(x > 3) ∨ (x < -3) ∨ (x > 3)`: `(x < -3) ∨ (x > 3)``
`+`
`+### Problem 6`
`+`
`+1. Either π is greater than 3, or it is greater than 10.`
`+2. x does not equal 0.`
`+3. x is greater than or equal to 0.`
`+4. x is greater than or equal to 0 (the `(x > 0)` clause is redundant here).`
`+5. Either x is less than -3, or it is greater than 3.`
`+`
`+### Problem 7`
`+`
`+Attempt to show one of the pieces is true, starting with the one most likely to`
`+be true (which will let you short-circuit the rest).`
`+`
`+### Problem 8`
`+`
`+Attempt to show one of the pieces is true, starting with the one most likely to`
`+be true (which will let you short-circuit the rest).`
`+`
`+### Problem 9`
`+`
`+1. `¬(π > 3.2)`: `x ≤ 3.2``
`+2. `¬(x < 0)`: `x ≥ 0``
`+3. `¬(x^2 > 0)`: `¬((x < 0) ∨ (x > 0))`: `¬(x ≠ 0)`: `x = 0` (assuming we're`
`+   talking about ℝ)`
`+4. `¬(x = 1)`: `x ≠ 1``
`+5. `¬¬ψ`: `ψ``
`+`
`+### Problem 10`
`+`
`+1. x is less than or equal to 3.2.`
`+2. x is greater than or equal to 0.`
`+3. x equals 0.`
`+4. x does not equal 1.`
`+5. ψ is true.`
`+`
`+### Problem 11`
`+`
`+D = "The dollar is strong"`
`+`
`+Y = "The Yuan is strong"`
`+`
`+T = "New US-China trade agreement signed"`
`+`
`+1. "Dollar and Yuan both strong": `D ∧ Y`.  This one translates pretty easily.`
`+2. "Trade agreement fails on news of weak Dollar": `¬D ∧ ¬T`.  The loses the`
`+   "weak dollar *caused* the failure" aspect of the headline.  Maybe`
`+   `(¬D → ¬T) ∧ ¬D` would be better?`
`+3. "Dollar weak but Yuan strong, following new trade agreement": `T ∧ ¬D ∧ Y`.`
`+   Again, this loses the causation aspect.`
`+4. "Strong Dollar means a weak Yuan": `D → ¬Y`.  If this headline is talking`
`+   about something that is actually the case, and not a hypothetical, then`
`+   `D ∧ (D → ¬Y)` is probably more accurate.`
`+5. "Yuan weak despite new trade agreement, but Dollar remains strong":`
`+   `¬Y ∧ T ∧ D`.  This is hard, because it loses the "despite" part of the`
`+   headline.  I don't think there's a way around that though since there's not`
`+   "we expected" logical operator..`
`+6. "Dollar and Yuan can’t both be strong at same time": `(¬D ∧ ¬Y) ∨ (D ∧ ¬Y)`
`+   ∨ (¬D ∧ Y)` or maybe `¬(D ∧ Y)` or maybe `(D → ¬Y) ∧ (Y → ¬D)`.`
`+7. "If new trade agreement is signed, Dollar and Yuan can’t both remain strong":`
`+   `T → ¬(D ∧ Y)` or `T → ((D → ¬Y) ∧ (Y → ¬D))`.`
`+8. "New trade agreement does not prevent fall in Dollar and Yuan":`
`+   `T ∧ (D ∨ ¬D) ∧ (Y ∨ ¬Y)` or `¬(T → D) ∧ ¬(T → Y)`.  This one is tricky but`
`+   I think that last one is clearest.`
`+9. "US–China trade agreement fails but both currencies remain strong":`
`+   `¬T ∧ (D ∧ Y)`.  This one is pretty simple, but again loses that "it's not`
`+   what we expected" vibe.`
`+10. "New trade agreement will be good for one side, but no one knows which.":`
`+   `(T → D) ∨ (T → Y)`.  The headline is ambiguous, but they probably mean "good`
`+   for one side at the expense of the other", in which case:`
`+   `(T → ((D ∧ ¬Y) ∨ (Y ∧ ¬D)))`.`
`+`
`+To Think About`
`+--------------`
`+`
`+1. It's all semantics.  In the US "innocent until proven guilty" means that`
`+   until you're proven guilty, you're "innocent".  So in that case yes, `¬guilty`
`+   = innocent`.  But if we're just talking about purely whether they did it or`
`+   not (disregarding courtrooms), then still yes, `¬guilty = innocent` because`
`+   if they didn't do it, they didn't do it.`
`+`
`+   The only place where there's ambiguity is when you take meaning 1 of "guilty"`
`+   (proven guilty in a court of law) with meaning 2 of innocent (actually did`
`+   not do the crime) or vice versa.  Sure, when you mix meanings like that it's`
`+   obviously not going to make sense.`
`+`
`+2. The problem in this one is this line:`
`+`
`+   "In terms of formal negation, ["not displeased"] has the form ¬(¬pleased)..."`
`+`
`+   Which makes the assumption that "displeased" is equivalent to `¬pleased`.`
`+   That's invalid (`¬pleased` means "I did not have positive feelings about it"`
`+   while "displeased" means "I feel palpably negative about it").  It entirely`
`+   ignores the third case ("I had no overall feelings at all").`
`+`
`+   To actually capture what "not displeased" is trying to say:`
`+`
`+       let pleased mean "I had overall positive feelings"`
`+       let displeased mean "I had overall negative feelings"`
`+       let ambivalent mean "I had no overall feelings, positive or negative"`
`+`
`+       ¬displeased = pleased ∨ ambivalent`
`+`
`+`
`+`