+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`

+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

+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

+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.

+Similar to problem 4: start with the piece most likely to be false, which will

+let you short-circuit quickly.

+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)`

+1. Either π is greater than 3, or it is greater than 10.

+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.

+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).

+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).

+1. `¬(π > 3.2)`: `x ≤ 3.2`

+3. `¬(x^2 > 0)`: `¬((x < 0) ∨ (x > 0))`: `¬(x ≠ 0)`: `x = 0` (assuming we're

+1. x is less than or equal to 3.2.

+2. x is greater than or equal to 0.

+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)))`.

+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