Let A = {n ∈ Z; n is odd} and B = {n ∈ Z; 3n 2 + 7 is even}. Prove that A = B.

Q: Prove that for all integers m and n,if m and m + n are even,then n is even

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Q: Prove by induction that for all n E N, An C {r € Z : z is even}

A: Given a sequence, An of sets defined recursively by, A1=2, for all n≥2, An=y∈ℤ|y∈An-1 ∨ y-4∈An-1. To…

Q: 2. Prove that for any natural n the following holds 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n + 1

A: Note: according to our guidelines we can answer only second question  2)

Q: Let p be a prime. Show that there is exactly two elements a belongs to Zp s.t a^2 = 1.

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Q: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove, by induction, that n + 3n2 + 2n is divisible by 6, for n 2 0.

A: In this problem I have use mathematical induction method two times please read step by step.

Q: 2. Ifr + 1, show that for any positive integer n, a + ar + ar +...+ ar" = a(rn+1 – 1) - r - 1 the…

A: we need to prove if r≠1 thena+ar+ar2+.....+arn=arn+1-1r-1let pn=a+ar+ar2+.....+arnto prove…

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

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Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

Q: Prove ImA = A = Aln if A is m X n

A: Given: A be m*n matrix

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: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: Prove the following statements (a) Vn € N, (2i − 1)² = by mathematical induction: n(2n + 1)(2n − 1)…

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Q: Prove that if a and b are integers of the form 4n+1, then ab is of the form 4n+1.

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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: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

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Q: 14b. Prove by mathematical induction: Vn E N, n³ –n is a multiple of 3. -

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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: Let p be prime. Prove that if H ≤ G, |G| = 2p and H is not normal, then |H| = 2.

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Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: (f) Prove by induction that for any positive integer n, n(n +1)(2n + 1) j=1

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Q: Show that if n = p1p2⋯pk, where p1, p2,…, pk are distinct primes that satisfy pj − 1 | n − 1 for j =…

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Q: Prove that “if n3 + 5 is odd, then n is even” by using the following: a) Direct Proof b) Indirect…

A: If n3 +5 is odd then n is even. Given:  n3 +5 = 2K+1                (K is any integer)       n3  =…

Q: Let p e Z be prime. Prove that there do not exist nonzero a, b e Z+ such that a2 = pb2

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Q: 10.) For all nzt prove the following by matkematid. induction. + 22 32 hて 人 tt n 23 22 n+2 24

A: Given: Let P(n)=112+122+132+........+1n2≤2-1n Let Q(n)=121+222+323+........+n2n≤2-n+22nfor all n≥1

Q: Prove that for all integers n, if n² is even then n is even.

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Q: Let n E Z. Prove that n3 – 7n + 18 is divisible by 6 for every interger n. Do not use induction.

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Q: Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: Prove that, for all integers n > 1, 1+ 3+ 5+7+ ...+ 2n – 1 = n².

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Q: (1 + x)" > 1 + nx

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Q: Prove or disprove: there exists a E N such that for all n E N, an + 1 is prime.

A: According to the given information, it is required to prove or disprove:

Q: Prove that for any natural n the following holds 1 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n +1

A: We have to prove that for any natural n the following holds…

Q: Prove: If Am = O for some positive integer m, then I − A is nonsingular.

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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: Let n € Z. Prove that if 3 /n, then 3 | (2n² + 1).

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Q: Let p e Z be prime. Prove that there do not exist nonzero a, b e Z+ such that a2 = pb2.

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Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: Let a and b be positive integers and let d = gcd(a, b) and m =lcm(a, b). If t divides both a and b,…

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Q: Give a direct proof of the claim: Vn E N, if the last two digits of n are divisible by 4 then n is…

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Q: Let m >1 be an integer, and let R = Z/mZ. Prove that R is nice if and only m is prime. Proof. ...

A: Given: m&gt;1 is an integer and R=Z/mZ. To show R is nice if and only m is prime.

Q: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

Q: Prove: Vn e Z,if n is odd, then 3n+5 is even.

A: Consider the provided question, To prove: if n is odd integer, then 3n + 5 is even. Proof by…

Q: Suppose that j is prime and j2 | ab. Prove that if gcd(a, b) = 1, then j2 | a or p2 | b.

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Q: Let n e Z. Prove that if n is odd, then 5n + 13 is even.

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Q: Show that if A and B are both n × n, then (a) det(AT BT ) = det(A) det(BT ); (b) det(AT BT ) =…

A: Given A and B are n×n matrix.

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: Prove that ∀n ∈ Z, gcd(7n + 1, 15n + 2) = 1.

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