Define the sets A = {3n|n E N} and the sets B = {1,2,3,4} U {5n|n E N} Prove whether A < B

Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

Q: Let e be a positive real number. Prove or disprove that n E N(n²+e).

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Q: 14. Let (a, b) = 1 and 2 = y for some integers x and y. Then prove that there exists an integer n…

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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: How many k-lists {r1, x2, ..., xk} are possible, such that each x; is a positive integer and 1 <x1 <…

A: Here each xi is a positive integer and 1≤x1≤x2≤...≤xk≤n.

Q: Show that 1+2+3+… +n = (";").

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Q: Let a and b be integers such that b measures a. (a) Prove that –b measures a.

A: Given: b measures a To proof: -b measures a

Q: Prove: |P(A)|= 2" where |A| = n. (Here P(A) is the power set of the set A with n elements.)

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Q: prove that for all integers m, n & d if d/m & d/n then d/(2m+3n)

A: We know that if a and b are integers such that a/b then b is a multiple of a. i.e. b=ka where k∈ Z{…

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: 19. Let s andt be real numbers such that t - s > 1. Prove that there exists an integer p such that s…

A: Here we use the concept of greatest integer function. For a real number X, [X] denotes the largest…

Q: Match each set with an enumeration of the set. A. f(n) = 2n for every n EN Ev{nEN:n>1} B. f(n) =n+1…

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Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

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

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Q: Find the smallest integer in the given set. Let a, b, and e be integers such that a ≠ 0. Prove that…

A: Let a, b, and c be integers such that a ≠ 0. Prove that if a l bc, then a l c *(a, b). Assuming…

Q: rove that the given inequality holds for the specified integer values of n n^3>(n+1)^2, n≥3

A: we will solve this by induction on the inequality P(n)=n3&gt;(n+1)2, n≥3 here n≥3 so will first put…

Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: For each natural number n, let 1 n+1) = {r €R: <r< 2n+1} . An n2 Calculate (with proof) U An and N…

A: The objective is to calculate:

Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = {n €Z|n…

A: If y is an even integer then it will be of type 2n for some n in integer If y is an odd integer then…

Q: Let S be a set of integers defined as follows:   2 ∈ S if n ∈ S, then n2 ∈ S if n ∈ S, then 16n…

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Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: Prove by Mathematical Induction. Ei (i !) = (n + 1)! – 1, for all integers n 21

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Q: If x is a lower bound for S in R, then x must be unique. a. True b. c. Never True Sometimes True

A: We want to find the given statement is true of false.

Q: Let E = . Determine the smallest choice of N > 0 such that N. |n(n+1)|

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Q: (a) Show that if x E Q is such that |x|, < 1 for all prime numbers p then x E Z.

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Q: 59. Let N be the set of positive integers and for each k € N, let Ag = |0 ,2- Find U-1 Ak- =D1 А. 0…

A: Let ℕ be the set of positive integers and for each k∈ℕ, let Ak=0,2-1k

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: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: Let A and B be the following sets: A = {m ∈ Z | m = a^2 +3 for some odd integer a}, B = {n ∈ Z | n…

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Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: 5. Let E {en}n€N be a set of positive real numbers satisfying en+1 en+1 for all n e N. Prove that…

A: I will write 2 proofs. One which uses limit theorems and other by explicitly solving the recursive…

Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

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Q: Prove that the statements here are true for every positive integer n. a + (a + d) + (a + 2d) + · · ·…

A: To Determine: Prove that the statements here are true for every positive integer n. Given: we have…

Q: b) Prove, by induction, that for any k > 1 and any choice of c1, ... , Ck E R and X1, ... , Xk E V,…

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Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

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Q: Consider the following families of sets indexed by the set of natural numbers. i which among these…

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Q: 3. Consider the following definitions of some sets: U = {1,2,. 12} ACU where a E A if and only if a…

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Q: - Let n and r be positive integers such that r≤n. Then (a) P = nx pn-1 (b) P = P-1

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Q: B. Consider the function b(n) = 4n+3 for all positive integers n. Compute b(1), b(2), b(3), b(4),…

A: Sequence: An arrangement of the set of numbers or objects following a rule is called a sequence. We…

Q: (b) Let A = a + 1 and B = b+ 12. Using proof by induction, show that A+B divides An+1 + Bên+1, What…

A: here i used basic condition of divisibility and i used the notation a|b for a divides b .

Q: (a) Determine the set of values of n for which and are positive integers. A 'Pythagorean triple' is…

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Q: Prove: Let a,b,x and y be integers. If a|x and b|y then (ab)|(2xy).

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Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

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Q: Let k be any integer greater than 1. Prove that for all n 20, we je: (k- 1)-[ +k' + +.+*'] + 1="+1)…

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Q: Let A and B be the following sets: A = {m€Z|m=a² + 3 for some odd integer a }, B={n€Z|n is divisible…

A: Let, x belongs to A. Then, x = a²+3 for some odd integer a  So, a being odd, we can write a as…

Q: 4. Define sets A and B as follows: A = {n EZ In = 10r - 2 for some integer r} and B = (m €Z |m = 5s…

A: We have to check whether set A = B.