\fProof Methods 1. Direct proof 2.Dis proof using counterexample 3. Proof by cases 4. Indirect proof a. Proof by contradiction. b. Proof by contrapositio n. 5. Mathematical Induction. 15. State whether the statement is true or false, and either prove the statement or give a counterexample. a. The sum of two rational numbers is always rational. b. The sum of two irrational numbers is always irrational. c. The product of any rational number and any irrational number is irrational. d. Given integers a, b, and c* ifa | b and a|c, then a| {5b+3c:l. 16. If you were writing the proof of the following statementt for each of the methods below, write the beginning statement [Let...] and what you will need to prove. Then prove the statement using the best suited method. v integers rt, if n is odd then 1:2 is odd. a} Method of Contradiction b} M th d of ntra ition :1 Method of Direct Proof {1} Proof by (continue from above] 1?. Consider the following statement: For all integers In if n.3is even then n is even. [a] Prove the statement either by contradiction or by contra position. lClearly indicate which method you are using. [b] If you used proof by contradiction in part {a}, write what you would "suppose\" and what you would "show"r to prove the statement by contraposition. If you used proof by cont raposition. in part {a}, write what you would "suppose" and what you would "show" to prove the statement by contradiction. 13. Consider the following statement: For all real numbers ry if r3 is irrational then r is irrational. {a} Prove the statement either by contradiction or by contraposition. Clearly indicate which method you are using. [b] If you used proof by contradiction in part {a}, write what you would "suppose\" and what you would "show"r to prove the statement by contraposition. If you used proof by cont raposition in part {aL write what you would "suppose" and what you would "show" to prove the statement by contradiction. 1. Write the following definition and theorem. Even Odd Rational number bla div& mod The quotient-remainder theorem 2. True or False? a) 412 b) 9 36 c) -7 151 d) 5 13 e) 13 10 3. a.32 div 9 = b. 32 mod 9 = c. Suppose a is an integer. If a mod 5=3 and b mod 5 =1, What is ab mod 5 ? d. Suppose x and y are integers. If x mod 7 = 3 and y mod 7 = 2 , what is ( 2x + y ) mod 7 = ? 4. Given that n = 153 compute the following remainders (show/explain work): a) n mod 3 , d div 3 b) n mod 5, n div 5 c) n mod 7 , n div 7 5. Express the integer 792 as a product of prime powers, in standard factored form. 6. Find the first five terms of sequences. by= 21-1 fori 2 0 7. Find a formula ( and give index range) : NIW 8. a. Evaluate: Ek=o(3k2 + 1) b. Evaluate: Ex-o (-1)*(2k -1)