Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y, if 5xy + 8 is even, then at least one of x or y must be even. 1.    Let x and y be integers. We will assume that it is not true that x or y is even and will show that 5xy + 8 is odd. 2) 2A) If it is not true that x or y is even, then x and y are both odd 2B) Therefore 5xy + 8 = 5(2k + 1)(2j+1) + 8 2C) x = 2k + 1 and y = 2j + 1 for some integers k and j 3) 3A) 5(2k + 1)(2j + 1) + 8 = 20jk + 10j + 10k + 5 + 8 = 2(10jk + 5j + 5k + 6) + 1 3B) Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer 3C) Therefore 5xy + 8 = 5(2k + 1)(2j + 1) + 8 3D) x = 2k + 1 and y = 2j + 1 for some integers k and j   4) 4A) 5(2k + 1)(2j + 1) + 8 = 20jk + 10j + 10k + 5 + 8 = 2(10jk + 5j + 5k + 6) + 1 4B) Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer 4C) Therefore 5xy + 8 = 5(2k + 1)(2j + 1) + 8 4D) x = 2k + 1 and y = 2j + 1 for some integers k and j   5) 5A) 5(2k + 1)(2j + 1) + 8 = 20jk + 10j + 10k + 5 + 8 = 2(10jk + 5j + 5k + 6) + 1 5B) Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer 5C) Therefore 5xy + 8 = 5(2k + 1)(2j + 1) + 8 5D) x = 2k + 1 and y = 2j + 1 for some integers k and j

Q: Give a proof by contradiction of the following: “If x and y are even integers, then xy is even”.…

A: To prove the statement "If x and y are even integers, then xy is even" by contradiction, we will…

Q: Use a proof by contradiction or by cases to show: No integers y and z exist for which 2y2 + z2 = 14

A: Result: Product of two positive term is always positive and product of two negative term is always…

Q: Prove: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

A:

Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

A:

Q: Prove, using the definition of odd, that for all integers r, that 6r+5 is odd.  b) true or false:…

A:

Q: lowest terms (that is, a and b have highest common factor 1), then b/2a is in S. (iii) If a/b and…

A: First, let's prove that all elements in S lie in the interval [1/2, 1].1. By rule (i), 1/1 is in S.…

Q: For all integers 1. x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5.

A:

Q: How many triples (x1, 2, 13) of nonnegative integers are there where Xi + x2 + x3< 11. Suggestion.…

A: From the given information. The given inequality is as follows. The suggestion given is to convert…

Q: a

A:

Q: Let a and b be two integers. Consider the following statement: If a+b= ab, then a and b must both be…

A:

Q: Let x and y be two real numbers such that x + y is rational. Prove by contrapositive that if x is…

A: Introduction: Rational and irrational numbers are two very important divisions of real number…

Q: If r and s are real numbers and r < s, then there exists a positive integer t such that r+t = s.…

A:

Q: Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and…

A:

Q: 8. Show that for all integers n, n² – 3n is even. Give a full proof supporting each statement you…

A: Below are the steps used to prove any mathematical statement by the principle of induction. Prove…

Q: 3. Let x be an integer. Use the previous result to prove the following corollaries. (a) If x divides…

A: Divisibility: If a and b are integers, we say that a divides b, and write a|b, if there is an…

Q: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

A:

Q: Prove the following by using: a contrapositive argument; the fact that a sum of integers is an…

A:

Q: 5 points) Use a proof of the contrapositive to prove the following statem For all integers a, b, if…

A: The contrapositive of a statements  " If p then q  '' is    " If not q then not p "

Q: Here are two conjectures: Conjecture 1: For all integers a,b and c, if a|b and a|c, then a|bc.…

A:

Q: Use proof by contradiction to show that no integers  y and z exist for which 24y + 12z = 1.

A:

Q: Which proof method is used in above proof. O Proof by counterexample O Direct proof O Proof by…

A:

Q: Use a proof by contrapositive to prove that, if (x + 12)/x is an integer, then x < 13.

A:

Q: Let a, b, c be positive integers such that a is a multiple of c and a2 + b2 is not a multiple of c.…

A: a, b, c are positive integers such that a is a multiple of c.  =&gt; there exists some integer x…

Q: Write a proof by contradiction of the following. Let x and y be integers. If x and y satisfy the…

A:

Q: Assume the following three statements are true. p: For all real numbers x and y, if xy is…

A: We are given that statements p,  q and r are true. We need to prove that 2a-1 or a2+3 is…

Q: Let n, m, a, b be positive integers. motl Lof oontredietion u th

A: Given, n, m, a, b be positive integers. To prove that…

Q: X7. We know that given any real numbers x1 and x2, if x1X2 0, then x1 0 or x2 = 0. Since there are…

A: Given statement is P2 where P2: for any two given real numbers x1 and x2, if x1x2=0, then x1=0 or…

Q: 8. For the following statements, either prove that they are true or provide a counterexample: (a)…

A: Given that, For all integers a, b and c,  a|c and b|c Definition of divisibility: If a|c then there…

Q: Determine (with proof) if there exist integers m and x such that 1997m + 47x2 = 3(17x – 1). | If…

A: Given equation is 1997m+47x2=317x-1. To determine: Whether integers m and x exists or not.

Q: Prove that n is odd if and only if 3n+9 is even. You must use definitions of even (and odd, if…

A: An even number is an integer that is divisible by 2, i.e., when divided by 2, the remainder is 0.An…

Q: Use a combination of techniques such as proof by contrapositive and proof by cases to prove the…

A: The contrapositive statement is : "If x is an integer, then (x³ - x) is even" (1) Assumption : we…

Q: Prove the following statement using a direct proof. If x and y are rational numbers, then 4x + y is…

A:

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

A:

Q: Let n be an integer, use the definition of even to prove that 10n+ 4 is even. 10n+4=0x0 Since is a…

A: Let n be an integer. To prove that for any integer n the number 10n+4 is even. An even integer is…

Q: Let x be an integer. Prove or provide a counterexample: If 4x+7 is odd, then x is odd.

A: We provide a direct proof and give counter example to show that given statement is false.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.