Write a proof that proves the Union Un an element of natural numbers (-1/n, 2-1/n)=(-1, 2). Also write a proof that proves the intersectionn an element of natural numbers (-1/n, 2-1/n)=[0,1].

Q: Find all solutions to the system of congruences x ≡ 1 (mod 2), x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡…

A:

Q: 11 Plot the points (2, 0), (2,-3 (2, 4). (2, 1), and (2,-1) Describe the collection of all points of…

A: topic - coordinate geometry

Q: does not exist

A: Given that:  The limit   lim(x,y)→(0, 0 ) x2 + yx2 + y2   .

Q: Without using any theorems from the book, prove the following statement: The sum of any two odd…

A:

Q: 6) find all the and non residues (mods) incongruent quadratic residues

A: It is given that Incongurant quadratic residues.

Q: A finite sequence of numbers is called AWESOME if each subsequent member of the sequence, except the…

A: Given: To determine the natural number of x and y.

Q: Find all three-digit numbers with different digits with the property that those numbers are…

A: Given that it is to be found all three-digit numbers with different digits with the property that…

Q: Use mathematical induction to prove that +1+2+4+...+2"-² = 2"- for all 2 -- positive integers . п.

A: We need to prove using mathematical induction given statement First we will show statement holds for…

Q: Prove that for all integers, n , n^3-n is divisible by 3.

A:

Q: Solve the inequality. Give the resu lt in set notation. 2(x + 3) + 4x - 4 < 8x

A: The given equation is  2(x +3) +4x - 4 &lt; 8x

Q: The diagram below shows ten circles joined by five lines with three circles Place all the…

A: Requirements: Numbers should range from 1 to 10.   Every line has three circles.   3. The amount for…

Q: Outline a proof of the theorem by contradiction

A: On considering the theorem If A and B are comingled matrices then the product AB is commingled…

Q: 0 (HINT: First move terms around so that all terns are positive)

A: To prove:             ∑k = 0n(-1)k nk = 0

Q: -, 1+2+2 '+.+ 2 -1 for all natural numbers n.

A: We will use the principle of mathematical induction to prove the result.

Q: Determine whether the statement is true or false. Justify your answer with a proof by Direct Method…

A: The product of any two integers is odd. - True

Q: Give an example of numbers x, y such that |x| + |y| = x − y.

A: Given that: give an example of numbers x, y such that |x| + |y| = x − y.

Q: A finite sequence of numbers is called AWESOME if each subsequent member of the sequence, except the…

A: Find the product of already given natural numbers. To make the product of all four natural numbers a…

Q: Find integers x and y such that 3220x + 6006y = gcd(3220,6006).

A:

Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Prove that all generators of Z20 are prime numbers. Please provide complete solutions. Thank you.

A: The statement is incorrect as 9 is generator of Z20, but 9 is not prime number.

Q: Let n and k be positive integers with 2k + 1 ≤ n. Find the number of ways to choose k children from…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: 9. Prove that there are as many positive real numbers as there are real numbers.

A: Consider the set X and Y. Then we said that |X| = |Y| if there exists bijective function from X to…

Q: Use a graph to find a number 8 such that if x- < 8 then Itan x - 1|< 0.2.

A: According to the given problem, tanx-1&lt;0.2 Apply the property of modulus, ⇒-0.2&lt;tanx-1&lt;0.2…

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: Define Conjugate transpose.

A:

Q: How many bijective functions have [10] as both the domain and codomain

A: Given, 10

Q: Prove that AXB|=A²B² - (AB)².

A:

Q: Prove that the product of two odd numbers is odd.

A: We first define a general odd number. Lets say that n is an integer. When we multiply any number by…

Q: (4) Prove that there is no smallest positive real number.

A: I have to prove that there is no smallest positive real number . Suppose that there is a smallest…

Q: Carry out all the steps in the the division algorithm for 78 and 15. This means show how to find q…

A:

Q: An operation * is defined on the set R by x * y = x +y+ 6 where x, y are elements of R. What is…

A:

Q: Prove a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

A:

Q: Let x and y be non-negative real numbers. Arithmetic mean of x and y is defined by 2, and the…

A: The problem is related to descriptive statistics. Under statistics the arithmetic and geometric mean…

Q: Select all the rules for the translation of RST.

A: From the figure we can clearly see that the co-ordinates of  the points of triangle RST and triangle…

Q: 2. Find all Pythagorcan Triples (not just primitive ones), containing the integer 15.

A: The Pythagorean triples are positive integers a,b,c such that a2+b2=c2. We can say that the…

Q: How many paths are there from point (0,0) to point (50,50) on the two-dimensional plane if each step…

A: Look at the following diagram: (Note: This is just for illustration and NOT drawn to the actual…

Q: Prove that (1-x*)у ,-ху, +р'у-0

A:

Q: Show that if x is an integer, then gcd (x+1, x^2 -x +1) divides 3. Use this result to find knowing…

A: Solving

Q: 3. Show that (5+ 13)e + (5- i3)-" is a real number if is also a real number.

A:

Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

A:

Q: Which of these relations on {0, 1, 2, 3} are equivalence ?relations

A: Equivalence relations are the relations which are reflexive (xRx), symmetric (xRy implies yRx),…

Q: Use Roster Method to represent the sets of natural number that satisfy x + 3 ≤ 7

A: We know that N={1,2,3,....}. And natural number less than equal to four are 1,2,3,4. We use these…

Q: Two consecutive natural numbers are given such that the greater of those number is a perfect square.…

A: Here I am attaching image so that you understand each and every step.

Q: Prove that the simple continued fraction of a real number is unique i.e. if [a0,a1,...] = x =…

A: To Prove: The simple continued fraction representation of a real number is unique, i.e. if…

Q: Assume that 2n tennis player play 2n -1 days in a tournament, playing each day one match against a…

A:

Q: Exercise 1.8. Prove that if one chooses n+1 numbers from {1, 2, 3,...,2n}, it is guaranteed that two…

A: Prove that if one chooses n+1 numbers from the set {1,2,3,…,2n}, then two numbers will be…

Q: Give an example of a pair of numbers that can be used to generate modulo residues to represent the…

A:

Q: How many paths are there from point (0,0) to point (50,50) on the two-dimensional plane if each step…

A: Look at the following diagram: (Note: This is just for illustration and NOT drawn to the actual…

Q: If {an} diverges and {bn} diverges, then {an + bn} diverges.

A: we have  to check If {an} diverges and {bn} diverges, then {an + bn} diverges.

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.