Question

please answer questions #7-13

7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n + 2 is even, then n is even using proof by contradiction. 11. Prove that for every n E Z,n2 2 is not divisible by 4. Use proof by Contradiction. 12. Prove that if a and b are consecutive integers then the sum a + b is odd. Use proof by contrapositive. 13. Let m,n 2 0 be integers. Prove that if m+n 229 then m 2 15 or n 215. Prove by contradiction.

Answer 1

7)

Every odd integer is the difference of two squares. Recall that process of direct proof. A direct proof shows that a conditional statement p → q is true, by showing that, if p is true, then q must also be true

Let n be any odd integer, then there is any integer k such that n 2k1 Let any integers k,k+1 -k2 +2k+1-k - 2k +1 Since (a+ba+2ab +b Therefore,n (k+ -k* Thus, every odd integer is a difference of squares



8)


We disprove that the product of two irrational numbers is irrational. This means, the product of two irrational numbers is not necessarily irrational. To disprove this, we give a counter example.

Let a = V3 = b be a nonzero irrational number. Then a-b=./3-B -3 is rational number Thus the product of two irrational numbers is not



9)


The objective is to prove the statement "if x+y 2 2,where x and y are real numbers, then x21 or y21. by the method of contraposition Consider the statements as, p:xty22, here x and yare real numbers. g: xžlor y21 To prove p → q by contraposition, assume-q as a premise, and using axioms, definitions and previously proven theorems, together with rules of inference show that -p follows.


Suppose that x21 or y2lis not true Then, x<l and y <1 Add the above inequalities as follows, Thus, x+y+1-2 That is, x+y<2 This is a negation of x+y22. Thus,---p. Hence, by contraposition, if x+y22 where x and y are real numbers then x21 ory2



10)


Use contradiction method to prove that 3m+2 is even when n is an integer And then n is even. Suppose that 3n+2 is even and n is odd. Since nis odd and then product of two odd numbers is odd, it follows that 3n is odd and then that 3n +2 is odd Therefore, the assumption that 3n+2 is even and n is odd is wrong This is contradiction. Thus, if n is an integer and 3n+2 is even then n is even.





Dear student, we are allowed to answer max 4 questions ..Chegg policy
Feel free to comment below for any queries.
Please do rate the answer if it was helpful ...thanks...