If we want to prove by contradiction the statement: For any positive real number a there exists a natural number n such that the product na > 1, what should we assume? there exist positive x ER and n E N such that na <1. O for any positive a € R there exists n E N such that ne < 1. there exists a positive a €R such that for any n E N, na < 1. there exists a positive a ER such that for any n E N, nr <1.

Q: -. For any integer not an integer, then [x]+[m- x] = m-1. m and any real number x, if x is

A: Let m be any integer and x be any real number that is not an integer.

Q: (a) Prove that 3n2 +1 is even if and only if n is an odd integer. (b) Suppose A, B,C and D are sets.…

A: The given question has two parts. The solution is explained below.

Q: For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n +…

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Q: Suppose we have a proof with the following proof header: Let æ be an arbitrary natural number.…

A: * SOLUTION :-  From the above information  The OPTION B is correct answer.

Q: Theorem. Let n e Nand let a E Z. If a = ng +r and 0 <r<n for some integers q and r, then a =…

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Q: If we want to prove by contradiction the statement: For any positive real number x there exists a…

A: We have to choose the correct answer that we should assume to prove the given statement by…

Q: Let n € Z. Prove that (n + 1)² - 1 is even if and only if n is even.

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Q: . Problem 4: critize the following proof – find all mistakes(there may be more than one). A mistake…

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Q: Prove that A=I-N+N²-N³ for A & N definec as follows: N=A-I (identity), where |1 0 0 an & I= 0 1 0. 0…

A: given matrix  A=1a12a1301a23001 and I=100010001 prove  that A-1=I-N+N2-N3 where N=A-I

Q: 2: If a ,,a,,a,,.,a, are in AP with first term a, =0 and common difference d 0, then prove that 1 1…

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Q: If we want to prove by contradiction the statement: For any positive real number x there exists a…

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Q: Suppose E = { € R: x² + x ≤ 1}. Select the best answer. The supremum of E is a rational number but…

A: The given set E=x∈ℝ: x2+x≤1. We have to find the supremum of the set.

Q: Suppose n E Z. Prove that if n² + 3 is even, then n is odd.

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Q: Prove the following step by step with explanation please,

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Q: 3. Let h and k be integers such that h|k and k|h. Prove that |h| = |k|.

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Q: 2. Prove that if A, B,C are sets such that ANBNC = 0, then (A\B)n(C\B) = AnC. Prove that for onu got

A: Recall that, set A\B means a set of elements of A that does not belong to B.

Q: Suppose E = { € R : x² + x < 6}. Select the best answer. The set E is empty. The supremum of E is an…

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Q: Assume that an and bn are eventually the same, so there is some K EN such that an = bn for all n >…

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Q: Prove that if m | k then 2" – 1| 2k – 1

A: Use the definition of Divisibility, "a divides b" means a and b are integers and there is an integer…

Q: Let x be a real number. Prove the following statement: for each e > 0, there exists a rational…

A: Given  that x be a real number. And we have to prove the following statement: for each ε&gt;0, there…

Q: solve the differenctil equation by undtermeind coefficients y'' +2y = -18x2e2x   y(x) = ?

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Q: Prove that for each a ∈ R and each n ∈ N there exists a rational number rn such that |a − rn| < 1/n

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Q: Example 1.41. Both the set Q of rational mumbers and the set R - Q of irrational numbers are dense…

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Q: Prove that If k, q, w and r are integers, and if c|k and c|q, then c|(kw + qr)

A: We know that  Definition of Divisibility : If a and b are integers such that a≠0, then we say "a…

Q: Let k > 2 be an integer and let n1, n2, ... , Ns E N be pairwise coprime positive integers. Show…

A: We prove given statement by mathematical induction in 3 steps. Step 1: We prove given statement for…

Q: Give a direct proof of the following: Let x ∈ Z. If 11x − 5 is odd, then x is even.

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Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

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Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b € N, and odd c€ N such…

A: Since n is even and non-zero . Since n is non zero ⇒ n is either positive or negative. Let's write…

Q: Theorem 24. If ɛ > 0, then there is an integer N EN such that 0 < + < e. .

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Q: Suppose n1,n2 are positive integers and gcd(n1,n2) = 1. Then for every integer x, if n1 | x and n2 |…

A: Given that gcd(n1,n2)=1 Then there exist integers m1 and m2 such that          m1n1+m2n2=1 Then…

Q: Theorem 1.22 If a e R. be R and a <b then there is a rational number r such that a <r< b.

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Q: The number of houses that can be served by a water pipe varies directly as the square of the…

A: Given that, The number of houses that can be served by a water pipe varies directly as the square of…

Q: I. Prove the following statements using the indicated proof. 1.) Let r E Z. If r is odd, then (r…

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Q: Let n € Z. Prove that (n + 1)² – 1 is even if and only if n is even

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Q: An integer x is odd if x2 is odd. Let x, y and z be integers. If x y and y z, then x | z. For any…

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Q: Consider the following definitions. integer n is alphic if n = 4k+ 5 for some integer k. Definition.…

A: An integer n is said to be alphic if n=4k+5 for some integer k.  An integer n is said to be gammic…

Q: 1. In this exercise we give an alternate proof that Q+, the set of positive rational num- bers, is…

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Q: that arn<1/n.

A: According to Archimedean property, For each real number a, there exists always an integer k such…