(a) For integers a, b, q, r, prove that if a = bq + r then (a, b) = (b, r).

Q: _If A = Verify that [A-!]" = [A"]¬1 ? l1 2.

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Q: for any integer a, show that gcd(2a+1,9a+4)=1.

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Q: Let n EU = Z and p(n): n² + n is an even integer. a. Write the following statement: (Vn) p(n). b. Is…

A: (a) Here, the given statement is: For ∀n∈U=ℤ, p(n):n2+n is an even integer.

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Assume that P(A₁ N A₂ NA3) > 0. Prove that P(A₁ NA₂ NÃ3 Ñ A4) = P(A1)P(A2|A1₁)P(A3|A1 N A₂)P(A4|A₁…

A: We have conditional probabilitiesP() =       (i)P( =                       (ii)P( =…

Q: Show that 1 1 1 x2 y? z2 (y² – a²)(z² – a²)(z²? – y²) - - ,4

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Q: (c) Prove that r| +11 36 T-1 2 +36 for all real I.

A: We have to prove that the following inequality holds: x-11x2+36≤x+1136 for all real x. We will use…

Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: Let ā= (2, 5, 7) and 6 = (1, 3, 4). (a - b) × (a + b) = X (3,16,33) X

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Q: Verify that v2+ V5 is algebraic over Q.

A: Algebraic number :- An algebraic number is a number that is a root of a non-zero polynomial in one…

Q: 8. Let a, b and c be any positive real numbers. Prove that b² + c² a²+b² 2c a² + c² + 2b 2a > a+b+c.

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Q: Find the argument of 1+i ал with integer a,b b write your answer as a/b for Arg (1+i) =-

A: We have to find the argument of 1+i in the form of arg(1+i)=aπb with integer a, b.

Q: If P(A) = P(B) = P(AB), show that P(AB + AB) = 0 [AB = An B]. %3D

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Q: (a+b)(a+ba)(ao³ + bw) = a' +b*

A: The question is wrong. Ask your guide for correction.

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: 6. Simplify these algebraic expressions. a) 2m + 12n - 3m - Op+ 10p +m – n- 12p b) 5r+ 13t +…

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Q: (p v q) → r = O (p v r)^(¬q v r) O (-p v r)v(q v r) O (-p v r)v(¬q v r) O (-p v r)^(¬q v r)

A: We will find out the required value.

Q: iken that I+2=1 xpressyin terms of xand z.

A: A function defines a relationship between two sets, where one set is dependent on other set. Any…

Q: Find the string that has this generating function: a) (1− x)−3 b) (1+2x)−4 c) 1/(1 −3x)3

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Q: (b) Let A and Bị be independent; similarly, let A and B2 be independent. Show that in this case, A…

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Q: Show that A × B ≠ B × A, when A and B are nonempty, unless A = B

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Q: For integers x and y, prove that x+y+xy + 3 is odd if and only if x and y are both even.

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Q: (c) Prove that x - 11 |x| + 11 |x² + 36| 36 for all real r.

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Q: If a, b, c are non-zero real numbers, then bc ca ab ca ab bc vanishes, when ab bc ca ca 1.1 -+- 0…

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Q: 8. For the following statements, either prove that they are true or provide a counterexample: (a)…

A: Given that, For all integers a, b and c,  a|c and b|c Definition of divisibility: If a|c then there…

Q: For an integer t, prove that gcd(2t + 1, 9t + 4) = 1.

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Q: Let a, b, c be integers. Prove that if a|b and a|c, then a|(b +c).

A: Suppose a|b and a|c. Then there are integers m and n such thatb = am and c = an.Then b + c = am + an…

Q: Prove that {4a + 126 : a, b E Z} = {4c: c € Z}.

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Q: Q2/ prove [A] [A] = [I] if [2 4 -5 [A]= 2 7 1 3 4

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Q: .

A: Given, The matrix,

Q: 1- Prove that (a) |A× B|² = |A|2|B|? – (A · B)² (b) (Ax B)(C × D) = (A· C)(B· D) – (A · D)(B · C)…

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