Math221 Matltematics for om utel' cience 3-2022 Assi nment 1 1. Use substitution method to prove that ~(X = (Y = Z )) a ~((X A Y) = Z ) with the name of the law clearly stated. (14 marks) 2. Let p, q and r be simple statements. Determine whether the following compound statement is a tautology, contradiction or contingent statement: 0: A r)V~(q A (:3 => :0) (14 marks) 3. Determine whether the following statements are True or False. (i) {1,302 {192}; (ii) 9 E P({ 1, 2M; (iii) 9 e P({1,2}); (iv) 9 E P({1, 2}II; (V) 9 E {{9}}; (vi) 9 E53, 1. 2}; (Vii) {9} E P({1, 2D; (14 marks) 4. Prove by contradiction that if m2 2m 3 is even, then m is odd. (14 marks) 5. Use direct prove to prove that for any two positive numbers at and y Page 1 of 2 (14 marks) 6. Prove by Mathematical Induction that for all natural numbers n, 13 + 23 + 33 + ... + n3 = n2(n+ 1)2/4. (15 marks) 7. Translate the following statements into symbolic statements without using definition: (1) The sum of the squares of three odd integers is even. (ii) All natural numbers are integers. (iii) There is at least one soldier who does not wear any helmet. (15 marks)