Consider the set I = {ACN|A is infinite and N\A es It's infinite} For example, the set P = {n € N | is cousin} It's in I then Pis infinite, just like its complement . Show that I P(N).

Q: Letr be a real number and let k be an integer. We then define the binomial coefficient () by (3)…

A: The Binomial coefficient is defined as: rk=rr-1...r-k+1k! for k≥1  k is an integer and r is a real…

Q: HW6 Proue →1<a = | <aca

A: We have to prove that  For all a &gt; 1, we always have 1 &lt; a &lt; a2 .

Q: 6. Define the floor [x] of a real number x to be "a rounded down", in other words, the greatest…

A: A relation S defined as S = {(x, y) | [x] = [y]} on the set of real numbers R is given, which is an…

Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: 9. Let F be the function such that F (n) is the sum of the first n positive integers. Give a…

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Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: |Let A¡ = [i − 1, i] for all integers i = 1,2,3,4. Find the following. a. A₁. A₂- b. A₁ U A₂ U A3 U…

A: The given set Ai=i-1, i for all integers i=1, 2, 3, 4. We have to find, a) A1, A2, A3, A4 b)…

Q: Show that if P(A|B)> P(A), then P(B| A) > P(B).

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Q: How many one-to-one functions are there from a set with m elements to a set with n elements, where…

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Q: 11 |72.6

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Q: Let A be a subset of real numbers bounded from below. Then, the set-2A = {-2x: x E A} is bounded…

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

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Q: Suppose that x is N(15,4). Find the P(X>11).

A: The random variable X follows normal distribution. The population mean is 15. The population…

Q: 1. Create a flow proof of the following result: For every real number x, if (x − 3)² ≤ 0, then 10-…

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Q: Let S be the set of all real numbers except -1. Define * on S by a*b = a + b + a b. The solution of…

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Q: Activity 7.2. Even if we omit the negatives, the real numbers fail to satisfy the well-ordering…

A: Given that the set R+ of all positive real numbers fails to be well ordered.

Q: How many injective functions are there from {1, 2} to {a, b, c, d}?

A: Let and .We have to determine the number of injective functions from  to .

Q: A = {5,6} B = {5, 7 ,8} S = {n | n >= 3 is an odd integer} A relation R from AxB to S is defined…

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Q: 11) Let H < K < G. Prove that |G:H|= |G:K| · |K:H| (Do not assume G is finite).

A: Given : H≤K≤G We have to prove that G : H=G : K. K : H

Q: b) Evaluate UTM L{t*e- 6(t-2)}. TM

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Q: Let n and r be integers such that n > 0 and 0 ≤ r ≤n. Prove that (²) ₁ = n (²-1) by counting in two…

A: Given, n and r be integers such that n&gt;0 and 0&lt;r&lt;n.

Q: 6. Let f and g be two arithmetic functions with g(n) = Σan f(d). If g is multiplicative, then so is…

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Q: Let A = (x = R: 7 ≤x < 20} and B = {7, 18, 19, 20, 21). What is A - B? A. {x ER: (7<x< 18) (18 < x…

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Q: Exercise 9.3.9. To test whether the number n is a prime, you divide n all the integers 1, 2, 3,...…

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Q: Give an example of numbers a and b such that |a + b| &lt; |a|+|b|.

A: To give an example of numbers a and b such that

Q: 3. Let S be the congruent modulo 2 relation on Z,that is m S n→ 2|(m-n) . Show the relation

A: A relation is called symmetric if for any  if  then we have .

Q: let a and m be the integers with gcd(a,m)=1, what is the relation between ord m (a) and ord m (-a)

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Q: 1 1 1 (c) Find simple descriptions for each of the sets U • U [ ₁ - ² x + ¹) and № [² - ² + ¹) X- n…

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Q: 2. Almost! But there's nothing that maps to 0. In your notes at the top you've listed ℕ={1,2,3,…},…

A: Solution:  f : Z → N Consider f(x) = |x| Now, we see, f(-1) = |-1| = 1 and f(1) = |1| = 1 although…

Q: Define the Fibonacci numbers as Fo = 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1…

A: Let's prove the identity Fn+1Fn−1−Fn2=(−1)n for n≥1 using mathematical induction.Base Case (n =…

Q: 6. Based on what you know about the multiplicative principle, how many functions f:A- Bare there if…

A: Solution:

Q: Consider the given predicate functions. P(n): "n is a prime number " E(n): "n is an even number" P…

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