For this problem it will be helpful to recall that all interpretations in FOL are assumed to have a non-empty domain.

1. Is the sentence xy(x=yy=x) valid in FOL? Explain.

2. Is the sentence x y x = y valid in FOL? Explain.

3. Write a closed FOL sentence all of whose models have a domain of cardinality 1.

4. Write a closed FOL sentence all of whose models have a domain of cardinality at least 2.