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Problem 1: For each argument below write the corresponding argument form. Then determine whether the argument is valid. If the argument is valid provide a proof of validity using rules of inference. If the argument is not valid provide an appropriate construction. (Clearly state domains of variables and meaning of predicates.) (a) Every dog is a carrot. There is at least one dog. Every dog likes singing. Therefore, every carrot likes singing TEGE PAR (b) Some people like coffee. Every person does not like reading or does not like dancing. Some people do not like dancing and do not like coffee. Therefore, there are people who do not like reading and like coffee. cones NIVERSITE D'OTTAWA - CON LE DROIT D'AUTEUR OTTAWA - COPYRIGHTED AL Problem 2: Make sure to clearly state the assumptions and all steps in your proof. Every variable that appears in your solution (for the first time) must be paired with an English word equivalent to the existential or universal quantifier, i.e., should be accompanied by a word "every", "some", "all", etc. Prove that for every real number z if x is irrational then + + 2 is irrational. Problem 3: Make sure to clearly state the assumptions and all steps in your proof. Every variable that appears in your solution (for the first time) must be paired with an English word equivalent to the existential or universal quantifier, i.e., should be accompanied by a word "every", "some", "all", etc. Prove that for all integers a and b if there exists an integer r such that ax2 + b = 0(mod 5) then a = b[mod 5) or a = -b(mod 5) or b = 0(mod 5)