. Let a be a positive number. Prove that for each real number x there is an integer n such that na < x < (n + 1)a.

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

A:

Q: 1. Let y be a positive real number. Prove that for every n e N there exists a unique positive real…

A:

Q: Let a, n E N. Assume that n > 2. Show that if Va is rational, then it is an integer.

A:

Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

A:

Q: Find the value of k such that A is singular. 1 0 5 3 -1 0 2 k k =

A:

Q: 4. Let IR be the set of all real numbers, xER and n EN such that n 2 2. If (1+x)> 0, prove that Pn :…

A: Let ℝ be the set of all real numbers,  Let x∈ℝ and n∈ℕ such that n≥2 where ℕ is the set of natural…

Q: Find the value of k such that A is singular. 1 0 4 5 -1 0 3 3 k k =

A: Topic:- Matrix

Q: Prove that if x is a positive real number, then LV L]] = [V]

A: We have to prove the given equation for all positive real numbers x .

Q: Prove that for any integer a, 3 divides

A: To prove that for any integer a, 3 divides (a-3)(a+4)(a+2).

Q: Prove that for every real number b, where b is not equal to 0,= b.

A: Given

Q: Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for…

A:

Q: (e) For any nonzero real number z, prove Σ''-1 1-z"

A: The objective is to prove that for any nonzero real number x, ∑i=1nx-1xi=1-x-n

Q: Prove the following by proof by contradiction: At least one of the 5 real numbers a, b, c, d, e is…

A: We will use basic properties  of inequality to proof this result by contradiction

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

A:

Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

A:

Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

A:

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x+ 5y² is even

A: Suppose x and y are integers. Prove that if x and y are odd, then x+5y2 is even.

Q: Let a and b be positive even integers. Prove that (a, by 2

A: Here, a,b denotes the gcd of a and b. a2,b2 denotes the gcd of a2 and b2. Given that a and b are…

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r&gt;d/2 where x, y, z belongs to Rn So locus z…

Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n &gt; 0.

A: Let a be a non zero real number , where m and n are integers…

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

A:

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

A:

Q: Let x be a real number. Prove that x is an integer if and only if [0.891a] + [0.109æ] =

A: Solution: Definitions floor function: It takes real number x as an input and gives as output the…

Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: Prove, for all integers x, that if x2 is odd, then x is odd.

A: Result: Every odd integer n can be written as n=2k+1, where k is an integer. Every even integer n…

Q: Prove: Provided it exists, (A-1)¯ = A.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: x]+n x+n m m

A: By using definition of greatest integer function, For x∈ℝ we have, x=x+x, 0≤x&lt;1 where, x is…

Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Let a and b be any real numbers such that a 0. Prove that a < b.

A:

Q: Show that there is exactly one real number t such that eTt + te" 207 + V2021.

A: Let us take an equation: g(x)=t Where g(x) be any function of x In order to show existence of x such…

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: 4. Let R be the set of all real numbers, xER and n EN such that n > 2. If (1+x) >0, prove that Pn :…

A: Mathematical induction.

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

A:

Q: Prove that e" > x+ 1, for all positive real numbers a.

A:

Q: Show that for any real number x and a subset A of R, exactly one of the following holds: (a) x is an…

A:

Q: {p € Z :x | p}U{p € Z:y|p} C {p € Z :n| p}. : n

A:

Q: Prove that the number V7 lies between 1 and 2.

A:

Q: b. H = {x E G|x" = e} for a fixed positive integer n.

A: Given set in part B

Q: Find the value of k such that A is singular. 1 0 6 4 -1 0 6 2 k

A: Given. A is a 3X3 matrix. We need to find the value of k for which the matrix is singular. Singular…

Q: Find the value of k such that A is singular. 0 4 6 -1 0 4 2 k k =

A:

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: m above, prove that for every real number c>0, there is a 1/N < c.

A: In every real number c, there is integer N which is N &gt; c.

Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

A:

Q: Prove that if m is a positive integer and x is a real number, then [m2] = [=] + z++ + 2 + |x + т — 1…

A: