consider the following sets:
a = {x | x is alive}
f = {x | x is in france}
m = {x | x is a national monument}
w = {x | x is a woman}
which statements are true regarding elements of the given sets? check all that apply.
□ paris ∈ f.
□ the statue of liberty ∈ a.
□ the statue of liberty ∉ m.
□ abraham lincoln belongs to none of these sets.
□ the eiffel tower is in more than one of these sets.

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# Brief Explanations:
1. For "Paris $\in$ F": Set F is {x | x is in France}, and Paris is in France, so this is true.
2. For "The Statue of Liberty $\in$ A": Set A is {x | x is alive}, and the Statue of Liberty is not alive, so this is false.
3. For "The Statue of Liberty $
otin$ M": Set M is {x | x is a national monument}, and the Statue of Liberty is a national monument (so it should be in M, making this statement false? Wait, no—wait, the statement is "not in M". Wait, the Statue of Liberty is a national monument, so it is in M, so "The Statue of Liberty $
otin$ M" is false? Wait, no, let's recheck. Wait, the Statue of Liberty is a national monument (in the US), so it should be in M. So the statement "The Statue of Liberty $
otin$ M" is false? Wait, maybe I made a mistake. Wait, the problem says "check all that apply". Let's re-express each:
   - Paris is in France, so Paris $\in$ F: true.
   - The Statue of Liberty is not alive, so $
otin$ A: false.
   - The Statue of Liberty is a national monument, so it is in M, so "The Statue of Liberty $
otin$ M" is false? Wait, no—the statement is "The Statue of Liberty $
otin$ M"—so if it is in M, then this statement is false. Wait, maybe I messed up. Wait, let's check the fourth: Abraham Lincoln is dead, so not in A; not in France (F); not a national monument? Wait, no, Abraham Lincoln is a person, dead, so not in A (alive); not in F (in France? No); is he a national monument? No, he's a person. Wait, set W is {x | x is a woman}, he's a man, so not in W. So "Abraham Lincoln belongs to none of these sets": is that true? Let's see: A is alive (he's dead), F is in France (no), M is national monument (no, he's a person), W is woman (no). So yes, he belongs to none: so that statement is true? Wait, no—wait, "belongs to none of these sets"—so if he's not in A, F, M, W, then yes. So that statement is true? Wait, maybe I made a mistake earlier. Let's re-express each option:
   1. Paris $\in$ F: Paris is in France, so true.
   2. The Statue of Liberty $\in$ A: it's not alive, so false.
   3. The Statue of Liberty $
otin$ M: it is a national monument, so it is in M, so this statement is false (because it is in M, so "not in M" is false). Wait, no—the statement is "The Statue of Liberty $
otin$ M"—so if it is in M, then the statement is false. So this is not true.
   4. Abraham Lincoln belongs to none of these sets: A (alive? No, he's dead); F (in France? No); M (national monument? No, he's a person); W (woman? No). So yes, he belongs to none: true.
   5. The Eiffel Tower: is it alive? No (A: no); in France (F: yes); is it a national monument (M: yes, it's a national monument in France); is it a woman (W: no). So it is in F and M, so "in more than one of these sets" (F and M): so true.
   Wait, so let's re-evaluate:
   - Paris $\in$ F: true.
   - The Statue of Liberty $\in$ A: false.
   - The Statue of Liberty $
otin$ M: false (because it is in M).
   - Abraham Lincoln belongs to none: true (not in A, F, M, W).
   - The Eiffel Tower: in F (yes) and M (yes, national monument), so in more than one (F and M): true.
   Wait, but earlier I thought the third was false, but maybe I was wrong. Wait, let's confirm each set:
   - A: alive things.
   - F: things in France.
   - M: national monuments.
   - W: women.
   So:
   - Paris: in France (F), so $\in$ F: true.
   - Statue of Liberty: not alive (A: no); in France? No (in US), so F: no; is it a national monument (M: yes, in US, so $\in$ M); so "The Statue of Liberty $
otin$ M" is false (because it is in M).
   - Abraham Lincoln: dead (A: no); not in France (F: no); not a national monument (M: no, he's a person); not a woman (W: no): so belongs to none: true.
   - Eiffel Tower: in France (F: yes); is it a national monument (M: yes, in France); not alive (A: no); not a woman (W: no): so in F and M, so "in more than one" (two sets): true.
   So the true statements are:
   - Paris $\in$ F.
   - Abraham Lincoln belongs to none of these sets.
   - The Eiffel Tower is in more than one of these sets.
   Wait, but let's check the third option again: "The Statue of Liberty $
otin$ M"—since it is a national monument, it is in M, so the statement "not in M" is false, so we don't check that. The second option: "The Statue of Liberty $\in$ A"—it's not alive, so false. So the correct ones are:
   - Paris $\in$ F.
   - Abraham Lincoln belongs to none of these sets.
   - The Eiffel Tower is in more than one of these sets.
   Wait, but maybe I made a mistake with Abraham Lincoln. Let's see: set M is {x | x is a national monument}—Abraham Lincoln is a person, not a monument, so not in M. Set A: alive—he's dead, so not in A. Set F: in France—no. Set W: woman—no. So yes, he belongs to none. So that statement is true.
   The Eiffel Tower: in France (F) and is a national monument (M), so in two sets: so "in more than one" (more than one means at least two) is true.
   So the true statements are:
   - Paris $\in$ F.
   - Abraham Lincoln belongs to none of these sets.
   - The Eiffel Tower is in more than one of these sets.
   Wait, but let's check the third option again. The Statue of Liberty is a national monument, so it is in M, so "The Statue of Liberty $
otin$ M" is false. So that's not true. The second option: "The Statue of Liberty $\in$ A"—false. So the correct ones are the first, fourth, and fifth? Wait, no—fourth is "Abraham Lincoln belongs to none of these sets"—yes. Fifth: "The Eiffel Tower is in more than one of these sets"—yes (F and M). First: "Paris $\in$ F"—yes.
   Wait, maybe I was wrong about the third. Let's confirm: "The Statue of Liberty $
otin$ M"—since it is a national monument, it is in M, so the statement "not in M" is false. So we don't check that. So the true ones are:
   - Paris $\in$ F.
   - Abraham Lincoln belongs to none of these sets.
   - The Eiffel Tower is in more than one of these sets.
# Answer:
- Paris $\in$ F.
- Abraham Lincoln belongs to none of these sets.
- The Eiffel Tower is in more than one of these sets.

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