22. Consider the statement “For every real number r, if r² is irrational then r is irrational." a. Write what you would suppose and what you would need to show to prove this statement by contradiction. b. Write what you would suppose and what you would need to show to prove this statement by contraposition.

3. Find the negations of each of the following statements. Is the original statement true or false? Is the negation true or false? (a) Every real number is either positive or negative. (b) There exists a positive integer n such that n is even or n2 is odd.

Suppose that we wish to prove a statement of the form P Q by CONTRADICTION. What should be the assumption in the proof? A. ~ P B. ~ C. ~ Q and P D. P

Exercise 4. Consider the statement VEZ, if x² is even, then x is even. Recall that we say that an integer is even if it is a multiple of 2. (a) Give the contrapositive of the above statement. (b) Give the negation of the original statement. (c) Which of the following are true? Circle all that apply. original statement Contrapositive negation Justify your answer but no proof is required.

State the negation for each of the following statements. 2 is a rational number. I am rich but I am not happy. (a) (i) (ii) (b) Determine whether each of the statement below is true or false. (i) (ii) (iii) -2 +2>2 or 1–4>0. 2 is a prime number and -1>-2. If 81 is multiple of 9, then 2+3 = 6.

A. Can every proof by contraposition be easily rewritten as a proof by contradiction? Explain. B. Can every proof by contradiction easily be rewritten as a proof by contraposition? Explain.

Let S be the statement: The reciprocal of any irrational number is irrational. ( The reciprocal of a nonzero real number x is - (a) Which of the following is the contrapositive of S? If a real number is not irrational, then its reciprocal is not irrational. O If a real number is irrational, then its reciprocal is irrational. If the reciprocal of a real number is not irrational, then the number is not irrational. If the reciprocal of a real number is irrational, then the number is irrational. If the reciprocal of a real number is not irrational, then the number is irrational. O If the reciprocal of a real number is irrational, then the number is not irrational. Write a proof by contraposition for S as a free response. (Submit a file with a maximum size of 1 MB.) Choose File No file chosen This answer has not been graded yet. (b) Which of the following is a negation for S? O If a real number is not irrational, then its reciprocal is irrational. If a real number is irrational, then the…

9. Given the statement : The sum of two odd integers is even. (a) Express it as an implication.

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39. Write the negation and the contrapositive of the following statement: If x is an even integer or x> 17, then x is a multiple of 4 and x>5. that do not have the same