Problem 1E: Use a direct proof to show that the sum of two odd integers is even. Problem 2E: Use a direct proof to show that the sum of two even integers is even. Problem 3E: Show that the square of an even number is an even number using a direct proof. Problem 4E: Show that the additive inverse, or negative, of an even number is an even number using a direct... Problem 5E: Prove that ifm+n andn+p are even integers, wherem,n, andpare integers, thenm+p is even. What kind of... Problem 6E: Use a direct proof to show that the product of two odd numbers is odd. Problem 7E: Use a direct proof to show that every odd integer is the difference of two squares. [Hint: Find the... Problem 8E: Prove that ifnis a perfect square, thenn+2 is not a perfect square. Problem 9E: Use a proof by contradiction to prove that the sum of an irrational number and a rational number is... Problem 10E: Use a direct proof to show that the product of two rational numbers is rational. Problem 11E: Prove or disprove that the product of two irrational numbers is irrational. Problem 12E: Prove or disprove that the product of a nonzero rational number and an irrational number is... Problem 13E: Prove that ifxis irrational, then 1/xis irrational. Problem 14E: Prove that ifxis rational andx0 , then1/x is rational. Problem 15E: Prove that ifxis an irrational number andx0 , thenx is also irrational. Problem 16E: Prove that ifx,y, andzare integers andx+y+z is odd, then at least one ofx,y, andzis odd. Problem 17E: Use a proof by contraposition to show that ifx+y2 , wherexandyare real numbers, thenx1 ory1 . Problem 18E: Prove that ifmandnare integers andmnis even, thenmis even ornis even. Problem 19E: Show that ifnis an integer and n3+5 is odd, thennis even using a) a proof by contraposition. b) a... Problem 20E: Prove that ifnis an integer and3n+2 is even, thennis even using a) a proof by contraposition. b) a... Problem 21E: Prove the propositionP(0), whereP(n) is the proposition “Ifnis a positive integer greater than 1,n2n... Problem 22E: Prove the propositionP(1), whereP(n) is the proposition "Ifnis a positive integer, then What kind of... Problem 23E: LetP(n) be the proposition “Ifaandbare positive real numbers, then (a+b)nan+bn . Prove thatP(1) is... Problem 24E: Show that if you pick three socks from a drawer containing just blue socks and black, you must get... Problem 25E: Show that at least ten of any 64 days chosen must fall on the same day of the week. Problem 26E: Show that at least three of any 25 days chosen must fall in the same month of the year. Problem 27E: Use a proof by contradiction to show that there is no rational numberrfor whichr3+r+1=0 . [Hint:... Problem 28E: Prove that ifnis a positive integer, thennis even if and only if7n+4 is even. Problem 29E: Prove that ifnis a positive integer, thennis odd if and only if5n+6 is odd. Problem 30E: Prove that m2=n2 if and only ifm=n orm=n . Problem 31E: Prove or disprove that ifmandnare integers such thatmn=1 , then eitherm=1 andn=1 , or elsem=1 and... Problem 32E: Show that these three statements are equivalent, whereaandbare real numbers: (i)ais less thanb, (ii)... Problem 33E: Show that these statements about the integerxare equivalent: (i)x+2 is even, (it)x+5 is odd, (iii)x2... Problem 34E: Show that these statements about the real numberxare equivalent: (i)xis rational, (ii)x/2 is a... Problem 35E: Show that these statements about the real numberxare equivalent: (i)xis irrational, (ii) x+2 is a... Problem 36E: Is this reasoning for finding the solutions of the equation2x21=x correct?(1)2x21=x is... Problem 37E: Is this reasoning for finding the solutions ofx+3=3x correct?(1)x+3=3x is given;(2)x+3=x26x+9 ,... Problem 38E: Show that the propositionsp1,p2,p3, andp4can be shown to be equivalent by showing thatp1p4,p2p3 ,... Problem 39E: Show that the propositionsp1,p2,p3,p4, andp5can be shown to be equivalent by proving that the... Problem 40E: Find a counterexample to the statement that every positive integer can be written as the sum of the... Problem 41E: Prove that at least real numbersa1,a2,…,anis greater than or equal to the average of these numbers.... Problem 42E: Use Exercise 41 to show that if the first 10 positive integers are placed around a circle, in any... Problem 43E: Prove that ifnis an integer, these four statements are equivalent: (i)nis even, (ii)n+1 is odd,... Problem 44E: Prove that these four statements about the integernare equivalent: (i)n2 is odd, (ii)1n is even,... format_list_bulleted