Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. The domain is all people on...

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. The domain is all people on Earth.

a) No one is perfect.

b) Not everyone is perfect.

c) All your friends are perfect.

d) At least one of your friends is perfect.

e) Everyone is your friend and is perfect.

f) Not everybody is your friend or someone is not perfect.

Express each of these statements using logical operators, predicates, and quantifiers.

a) Some propositions are tautologies.

b) The negation of a contradiction is a tautology.

c) The disjunction of two contingencies can be a tautology.

d) The conjunction of two tautologies is a tautology.