MATH221: Mathematics for Computer Science | My Assignment Tutor

MATH221: Mathematics for Computer ScienceTutorial Sheet Week 4Validating arguments1. Rewrite each of the following arguments into logical form, then use a truth table to checkthe validity of the argument.(i) If I go to the movies, then I will carry my phone or my 3D glasses. I am carrying myphone but not my 3D glasses. Therefore, I will go to the movies.(ii) I will buy a new bike or a used car. If I buy both a new bike and a used car, I willneed a loan. I bought a used car and I didn’t need a loan. Therefore, I didn’t buy anew bike.Methods of Proof2. Each of the following demonstrates either the Rule of Modus Ponens or the Law ofSyllogism. In each case, answer the question or complete the sentence and indicate which ofthe two logical rules is demonstrated.(i) If Caz is unsure of an address then she will phone. Caz is unsure of Graham’s address.What does she do?(ii) If x2 – 3x + 2 = 0, then (x – 2)(x – 1) = 0. If (x – 2)(x – 1) = 0 then x – 2 = 0 orx-1 = 0. If x-2 = 0 or x-1 = 0, then x = 2 or x = 1. Therefore, if x2 -3x + 2 = 0,then …(iii) We know that if x is a real number, then its square is positive or zero. So when py isa real number, what do we know about y?3. Prove or disprove the following statements.(i) For all natural numbers n, the expression n2 + n + 29 is prime.(ii) 9 x 2 Q s.t. 8 y 2 Q; xy 6= 1.(iii) 8 a; b 2 R; (a + b)2 = a2 + b2.(iv) The average of any two odd integers is odd.4. Find the mistakes in the following proofs.(i) Result: 8 k 2 Z; (k > 0 ) k2 + 2k + 1 is not prime).Proof: For k = 2, k2 + 2k + 1 = 9, which is not prime. Therefore, the result is true.(ii) Result: The difference between any odd integer and any even integer is odd.Proof: Let n be any odd integer, and m any even integer. By definition of odd,n = 2k + 1 for some k 2 Z, and by definition of even, n = 2k for some k 2 Z. Thenn – m = (2k + 1) – 2k = 1. But 1 is odd. Therefore, the result holds.15. Prove each of the following results using a direct proof.(i) 8 x 2 R; x2 + 1 ≥ 2x.(ii) For any n 2 N, if n is odd, then n2 is odd.(iii) The sum of any two odd integers is even.(iv) If the sum of two angles of a triangle is equal to the third angle, then the triangle is aright triangle.6. Prove each of the following statements using a proof by contradiction.(i) If n2 is odd then n is odd.(ii) There is no smallest positive real number.7. Prove each of the following statements using a proof by cases.(i) If x 2 f4; 5; 6g, then x2 – 3x + 21 6= x.(ii) 8 x 2 Z; x 6= 0 ) 2x + 3 6= 4.2

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