Prove. If x is a negative real number, there exists an integer k such that k >= x.

Q: Let n be a real number and let e be a positive real number. Determine whether there exists a real…

A: see the attachments

Q: Suppose that the number 'a' has a property that for every natural number n, a ≤ 1/n. Prove that a ≤…

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Q: Let n and k be natural numbers with n >k. how many subsets of {1,2, ..., 2n} have exactly k even…

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

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Q: 1. Prove that for every real number x, if |x-3| > 3 then x² > 6x. (Hint: Consider two cases)

A: Solution

Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that xn > y

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Q: List the members of each sets. 1( x/x is a real numbers such as x^3= 27) = 2. (x/x is an integer…

A: We will find each elements of the set

Q: Prove or disprove if x2-1 is odd, then x is even

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Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.…

Q: Prove that for every real number x, if x > 1 then there is a real number y such that y+ý = 2x.

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Q: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

Q: Let a and b be natural numbers, where a,b >= 2, such that gcd(a,b)=1 and ab=(x+y)n for some natural…

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

Q: Use the method of direct proof to prove the  statement If x is an odd integer, then x3 is odd.

A: To prove: If x is an odd integer then x3 is odd.

Q: Prove that if æ is a real number, then [2a] = [x] +

A: If x is a real number then  to prove- 2x=x+x+12 Greatest integer function is denoted by [x] for any…

Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: 10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely many rational…

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Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

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Q: Suppose that a is a positive real number not equal to 1, and that r and s are positive real numbers.…

A: To prove that; rlogαs=slogαr  We have to show that the left hand side of the equation equal to the…

Q: 60. Prove that for any positive integer n, for every real number x. %3D

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Q: Let a be a positive real number and let n be an integer greater than 1.Prove or disprove that…

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Q: If α is even, prove that α-1 is even. If α is odd, prove that α-1 is odd.

A: We know that the multiplication of 2-cycles can help us find any permutation of a finite set, so we…

Q: Prove that for all integers a, b, and c, if a2 + b2 = c2, then at least one of a and b is even.

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Q: If x and y are two positive real numbers, then there exists a positive integer n such that ny>x

A: Proof: Suppose there does not exist any positive integer n such that ny&gt;x then ny≤x   ∀n∈ℕ

Q: Prove that if x is a real number, then [2x] = [x] +

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Q: . Let x be a positive real number and d be a positive integer. Prove that the number of positive…

A: [.] is the greatest integer function.

Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that x" > y

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Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: 6. Suppose x is a real number for which x2 is irrational. Use a proof by contradiction to prove that…

A: Contradiction method means, If we want to prove If P implies Q, Then we are first assuming P and…

Q: a. Show that for any real number x, if x> 1 then | x4 |≤| 23x4 + 8x² + 4x |.

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Q: Prove that For every real number x > 0, there is a natural number M such that for all natural…

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Q: o Prove that for each real number x, (x + v2) is irrational or is irrational.

A: 8(a) Given x is real number. We know that 2 is a rational number. First we have to show that x+2 is…

Q: Suppose x is a real number. Prove that if x 1 then there is a real number y such that y+1 = x. 4-2

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Q: 3. Suppose a is an irrational positive real number. Prove that if x and y are integers and x # y…

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Q: Prove that if a and b are real numbers with a < b, then there are infinitely many rational numbers…

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Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: Let n>=2 and k be any integer, then prove that (n-1)^2 divides (n^k )-1 iff n-1 divides k.

A: See the attachment

Q: 4. Prove that for each real number x and each irrational number q, (x +q) is irrational or (x- q) is…

A: We prove by contradiction

Q: Show that |x^n| = |x| ^n for every positive integer n and every real number x.

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Q: Show that if a and b are integers such that a|b, then a^k|b^k for every positive integer k.

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Q: Suppose x is a real number and x is not equal to zero. Prove that if ((x^1/3)+5)/((x^2)+6)=1/x, then…

A: Given that, x is a nonzero real number, and 8 is not a root of given equation.

Q: Suppose x is a real number and x + 0. Prove that if then x 7 8. 2 16

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Q: Prove that for all real numbers x, if x is not an integer, then [r – 1] = [x] – 2.

A: We have to prove x−1=x−2 for all real number x, if x is not an integer. Use definition:- x is the…

Q: 11. Let y be a positive real number. Prove that for every ne N there exists a unique positive real…

A: This is a problem of real analysis.

Q: Prove that the statement is true for every positive integer n.

A: a&gt;1, then an&gt;1

Q: 2. Show that for any real number x, if x + 2 > 10, then x? – 14 > 5x. Use direct proof.

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Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…

Q: 3. Show that for any real number x, if x2 is odd, then x is odd.

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Q: 1 Prove that if z is a real number such that z+- is an integer, then for all natural numbers 1 n, z"…

A: Given that, if z is a real number such that: z+1z is an integer,Show that for all n,zn+1zn is an…

Statements and Logic

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