Translate each of these statements into logical expressions by using quatifiers and predicates with one or two variables. (a) A student in our discrete math class has lived in Florida. (b) There is a student in our discrete math class who got the perfect grade in Midterm I. (c) Everyone in our class loves discrete math. (d) There is a student in our class who has been to every state in the US. (e) There is a student in our class who has been to every city of at least one state in the country.

a) [tex]A = \{\exists x \in M, \exist y \in U\,|\,xGy \}[/tex], b) [tex]B = \{\exists x \in M,\,\exists y \in H\,|\,xMy \}[/tex], c) [tex]C = \{\forall x \in M\,|\,xI \}[/tex], d) [tex]D = \{\exists x \in M,\,\forall y \in U\,|\,xJy \}[/tex] , e) [tex]E = \{\exists x \in M, \forall y \in V, V \subseteq U\,|\,xJy \}[/tex]

Step-by-step explanation:

a) x - A student, M - Set of students of discrete math class, G - has lived in, y - Florida, U - Set of states of the United States of America.

[tex]A = \{\exists x \in M, \exist y \in U\,|\,xGy \}[/tex]

b) x - A student, M - Set of students of discrete math class, y - A perfect grade, H - Midterm I.

[tex]B = \{\exists x \in M,\,\exists y \in H\,|\,xMy \}[/tex]

c) x - A student, M - Set of students of discrete math class, I - loves discrete math.

[tex]C = \{\forall x \in M\,|\,xI \}[/tex]

d) x - A student, M - Set of students of discrete math class, J - has been in, y - a state, U - Set of states of the United States of America.

[tex]D = \{\exists x \in M,\,\forall y \in U\,|\,xJy \}[/tex]

e) x - A student, M - Set of students of discrete math class, J - has been in, y - a city, V - At least one state of the United States of America, U - Set of states of the United States of America.

[tex]E = \{\exists x \in M, \forall y \in V, V \subseteq U\,|\,xJy \}[/tex]