3. If |F|q<∞ show: (a) There exists a prime p such that char F = p. (b) q = p some n. (c) a = a for all a € F. (d) If b € K is algebraic over F then b9" = b for some m > 0.

Q: Ex: Let X₂ [1,2,3,4,5]. I. [0,X, [1], 82.3, 81.23], Y. [3,4,5,6,73 andl T₁: [0, X, [4], [4,633 if A:…

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Q: Supposer FCE, acE is algebraic F and dog (a/F) 13 odd. Proveo that F(a)= F(a²). over

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Q: . How many Boolean functions are there of degree 7? Explain.

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Q: d) Give a recursive definition of the sequence {a„}, n = 1, 2, 3... %3D n if a,= 2 + 1; and a15 e)…

A: d)  Here, given that an=2+1=3 So, for all n, an=3 which is the recursive definition.

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

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: Suppose that P(n) is a propositional function. Determine for which nonnegative integers n the…

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Q: (c) Prove that for n ≥ 1, n [27/7/2≤2- j² 1 n

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Q: Which set of points includes all of the solutions for y= 2/2

A: Giveny=52x+32

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: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: 3. Find all r so that r-1+r-2| < 5.

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Q: PAGE DATE FIND THE AREO IN THE FIRST QUOD ROINS BOUND ED BY FCX )= Yxx anD TiHE X-Oris.

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Q: 1. Let P (x) be the statement "x = x²." If the domain consists of the integers, give True/False…

A: We have given a statement ,  P(x) : x = x2 Here , domain consists of the integers , that is x ∈ ℤ.…

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

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Q: Let B, = {xER|0< x si for each integer i = 1, 2, 3, 4. (a) Find B, U B, U B3 U B4: (Enter your…

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Q: Prove:  C(n,k) = C(n-1,k-1)+C(n-1,k) for natural numbers k with 0<k<n.

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Q: Which of the following sets are countable? ORIN {7,99, 512}. 1 1 1 .}. " 10 100 1000 The set of all…

A: We know R is uncountable and N is coutable.  So, R \ N is uncountable [as R = (R \ N) U N, if R \N…

Q: Prove or disprove the following statement: "For all real numbers x, y, [xy] = [x][y]." (3 marks)

A: Introduction: Two important functions are frequently used in mathematics and computer programming.…

Q: a) 2x – 1 < x +3

A: Given inequality is  2x-1&lt;x+3

Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: (a) Suppose a, de N. Define what it means to say that d divides a. (b) Find all natural numbers d…

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Q: Suppose that P(n) is a propositional function. Determine for which nonnegative integers n the…

A: Sol:- We can use mathematical induction to prove that P(n) is true for all nonnegative integers n.…