More Symbolizations — 7.3ex IV

Symbolize the following.

universe of discourse: everything
g: George W. Bush
u: the United States
Cx: x is a country
Fx: x is female
Gx: x has a government
Hx: x is heroic
Px: x is a person
Yx: x is a capital city
Ixy: x is located in y
Lxy: x is the leader of y
Kxy: x knows y
(Please Note: the program which checks for logical equivalence between your answer and the expected one can "bog" down. These problems are at the limit of it's abilities! If the program is having difficulties with your answers, use the "hints" or just enter '?' and see if you can make sense of the expected answer. Then go back and try to do these on your own.)
  1. All countries have a government.
  2. No country has a government. (Hint)
  3. Some country has a govenment.
  4. Some country has no govenment.
  5. No country is led by George W. Bush.
  6. Every capital city is located in some country.(Hint)
  7. Some country has a capital city but no leader. (Hint)
  8. Some country has a capital city and is led by George W. Bush.
  9. If someone is the leader of the United States, then everyone knows that person. (Hint)
  10. Someone is the leader of the US, but no one knows that person. (Hint)