3. Prove that |x+ y| < |x| + \y] for all real numbers x and y.

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A: Since, we have to solve the given equation 16x-1=530x-5 -(i)

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A: (a) To prove: If x and y are rational then x+y is also rational. (b) To prove: If x is irrational…

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A: A polynomial P(x) is defined as

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A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

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Q: Discrete Math 2.4.3. Prove each of the following algebraic statements using a direct proof: a) For…

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Q: Question 1. If A is positive definite, show that A" is positive definite for all integers n > 1.

A: If A is positive definite

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A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

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A: Consider a matrix: A=2141-2126k

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A: The given statement is False.

Q: 9. True or False: “There is no way to assign truth values to the atomic components of an equivalent…

A: Clearly this statement is

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A: There are two possible answers for the given situation.If it is a single digit number, then it…

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A: See solution below

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A: We will solve this by using definition of logically equivalent

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A: Solution :-

Q: Prove: An odd plus an odd equals an even. (2m + 1) + (2n + 1) = [ ? ]m +[ ]n+ [ ] = 2(m + n + [ ) =…

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Q: Which of the following statement is always false? (a) For all integer x, [x] = [−x] (b) For all odd…

A: We have to check which of the given option is always false. a Given that  for all integers x,  x=−x.…

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Q: Identify the error or errors in this argument that supposedlyshows that if ∃xP(x)∧∃xQ(x)is true…

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Q: a) Given q(x, y) the numbers y+2, x-y and x+2y are even integers. (Domain discourse is a set of real…

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Q: 1. A = {x | x is a Prime number such that x <50 } 2. C = {x | x is a square of integer less than…

A: Both the questions are solved below. We will use the concepts of Sets.

Q: Problem 6. Determine all of the positive integers such that 3n < 2", and prove your result by…

A: Given statement is, Pn:3n&lt;2n Let, n=4 then, 34&lt;24⇒12&lt;16 So,  the statement is true for n=4

Q: a) By using mathematical induction, prove that the given equation is true for all positive integers.…

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Q: Problem 4: Mathematical Induction 7"+1 Prove that 2 + 2.7+2.72 +. . . + 2.7" = 1 whenever n is a…

A: We need to prove 2+2·7+2·72+....+2·7n=7n+1-13 When we prove by mathematical induction we need to…

Q: 5.45. Show that there exist no nonzero real numbers a and b such that va? + b? = Va³+6%. 5.46 Prove…

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Q: 5.46. Prove that there exists a unique real number solution to the equation x O between a 2/3 and r…

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Q: 1. Tell which property or axiom of the real numbers justifies each of the following mathematical…

A: 1)a) Given:              2x+y=2x+2y   Since here they are distributed through parentheses, it is a…

Q: If A is a set of real numbers that is not open, then A is closed.

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Q: For every real number x+1, n-1 1+x+x² +x' +. x´+x' +...+x" for all neN. 1-x

A: The question has a mistake , as in the R.H.S the power of x must be (n+1).

Q: Number Theory IS Prove: An odd plus an odd equals an even. (2m + 1) + (2n + 1) = [ ? ]m + [ ]n + [ ]…

A: Given that we need to prove  An odd plus an odd equals an even . Now, Let (2m + 1) and ( 2n + 1) are…

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Q: R.2. Prove that for every real number x, if x 0 and x +1 then there is a unique real number y such…

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Q: 6) The domain for variable x is the set of all integers. Select the statement that is true. a. ax…

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Q: Question 8: Using combinatorial argument, prove that (:) --C) - »C) * 311 +n* 3

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Q: Let n ≥ 1, x be a real number, and x ≥ −1. Prove the following statement using mathematical…

A: The principle of mathematical induction is a technique or method that is used to prove the statement…