QUESTION ONE (30 MARKS a) Determine the validity of the following argument S Some crazy people are dangerous S All fanatics are crazy Conclusion some fanatics are not dangerous n(n 1) 2 b) Prove the proposition that the sum of n positive even integers is c) Find all the integers such that 2 8 3n18 d) Give an example of a non trivial relation on the set A 1,2,3 which is i Both symmetric and antisymmetric ii Neither symmetric nor antisymmetric e) Solve the linear congruence equation 4x 6(mod10) (3 marks) (4 marks) (3 marks) (2 marks) (2 marks) (4 marks) f) Find the product of the polynomials f(x) 4x 2x 3x 1and g(x) 3x x 4 over Z5 (4 marks) (5 marks) g) Prove that (a b)' a b' h) Given a binary on the set of integers given by a b a b ab show that is (3 marks) commutative and associative

Question: QUESTION ONE (30 MARKS a) Determine the validity of the following argument S: Some crazy people are dangerous S: All fanatics are crazy Conclusion:

QUESTION ONE (30 MARKS a) Determine the validity of the following argument S: Some crazy people are dangerous S: All fanatics are crazy Conclusion:

QUESTION ONE (30 MARKS a) Determine the validity of the following argument S: Some crazy people are dangerous S: All fanatics are crazy Conclusion: some fanatics are not dangerous. n(n+1) 2 b) Prove the proposition that the sum of n positive even integers is. c) Find all the integers such that 2

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