Which of the following statements best justifies In(n+ 1)

Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

Q: Use the definitions  to show that the following statements are true. For example, to show that 5 |…

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Q: In (5 – x) – In (5 + x) = -In 9

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Q: 6) Apply induction to show that 2+4+ .2n n(n+1), n 21.

A: We prove this by the induction, firstly prove for n=1, then assume it is true for n=k, and then…

Q: The following is a valid identity, true or false? P(k): 1+2+22 +23 + .. +2k = (2(k+1)/2) +…

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Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: (26) Are there natural numbers n, m such that 20" = 50m? Justify your answer with a proof or a…

A: The idea is to use a prime factorization of the numbers 20 and 50 and then use this to solve the…

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: (c) If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # n m.

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Q: 7. Ug = kuz-1¬- Ug–2, for every integer k > 3 uj = 1, u2 = 1

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Q: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

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Q: 5 points) Use a proof of the contrapositive to prove the following statem For all integers a, b, if…

A: The contrapositive of a statements  " If p then q  '' is    " If not q then not p "

Q: 6) Let n be a positive integer such that 7 | (52n-1 +22n-1). Prove that 7 || (52n+1+22n+¹). (Hint:…

A: Since, given that  52n-1+ 22n-1  is divisible by 7 .  We have to prove that  52n+1+22n+1 is also…

Q: Suppose that P(n), Q(n) and R(n) are statements about the integer n. Let S(n) be the statement: "If…

A: Given that: P(n),  Q(n),   and   R(n) are the statements about n∈ℤ.   Also S(n) be a statement…

Q: Prove by induction that (−2)º + (−2)¹+ (−2)² + ... + (−2)¹ : = for all n positive odd integers.…

A: Let's start by plugging in n=1 into the formula given:   (-2)^0 + (-2)^1 = (1 - 2^(1+1))/3   1 +…

Q: Select the following statements that are true. gcd (2022, 111) = 1 2023 is a prime gcd (2022, 1685)…

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Q: Give a direct proof of the following statement. If x is an even integer, then x² – 6x + 5 is odd. |

A: Remark: We know that addition or subtraction of an even and an odd integer is always an odd integer.…

Q: Prove that n is odd if and only if 3n+9 is even. You must use definitions of even (and odd, if…

A: An even number is an integer that is divisible by 2, i.e., when divided by 2, the remainder is 0.An…

Q: 1.5 Inx - In(x2 - 1) + 2ln(x + 3)

A: Logarithm simplification

Q: Let P(n) be the statement n2 ≤ n! where n is a nonnegative integer. For which nonnegative integers…

A: check if the result is true for n=1 n2≤n!12≤1(true) for n=2 22≤2!4≤2(false) for n=3  32≤3!9≤6(false)…

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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Q: Suppose that P(1), P(2),... are statements. Assume that you know only the following: • P(3) is true.…

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Q: The following is a valid identity for the natural numbers, true or false? n(n +1)(n +2) P(n):13 +23…

A: The solution is given as

Q: (d))For all integers a, b, and c, if a | b or a c, then a | bc (e) For all integers a, b, and c, if…

A: (d) let for all integers a,b,c if a|b or a|c so a|b or a|c ⇔b = ak1 or  c = ak2  for some integers…

Q: Consider the following statement. There is an integer n such that 2n² - 5n + 2 is prime. To prove…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.