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Alisha Atif

Alisha A.

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Assume a unit circle. Consider a triangle whose edges/corners fall on the boundary of the circle (the three sides of the triangle are arcs of the circle). What is the probability triangle that the triangle covers the centre of the circle? provide a detailed reasoning for your answer

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Formally prove that log(n!) = O(nlogn)

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For each of the following pair of functions state with a brief justification whether F(x) is in O(g(x)) or not. (a) f(x) = 2^x , g(x) = 3^{x-1} (b) f(x)= x^{x^{2}} , g(x) = 2^{2^{x}}

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(lii) \( f: x ;=x^{3}, s(x)=z k z \) \( |c| ;|x ;=z, 2 x| \quad r \) : At \( \left\{t z ;=x^{\prime}, y, x \mid z^{\prime}\right. \)

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Task 3. Injective and Surjective Functions [6 points]For each of the following functions ( f ), determine if it is injective (one-to-one), surjective (onto), and bijective. In the latter case, provide the inverse ( f^{-1} ).(a) ( f: mathbb{R}^{2} rightarrow mathbb{R}^{2},(x, y) mapsto(2 y,-x) )

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Online Assignment #7 (iSavedNOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part Arrange the steps in the correct order to find an inverse of a modulo ( m ) for each of the following pairs of relatively prime integers using the Euclidean algorithm.Ch ( 04 mathrm{Sec} 4 ) Ex 05 (a) - Inverse of a modulo m using the Euclidean AlgorithmPart 1 of 2[a=4, m=9]5 pointsRank the options below.7 is an inverse of 4 modulo 9.The Bézout coefficients of 9 and 4 are 1 and -2 , respectively.The coefficient of 4 is -2 , which is the same as 7 modulo 9 .The steps to find ( operatorname{gcd}(4,9) ) using the Euclidean algorithm are listed below.[begin{array}{l}9=2 cdot 4+1 4=4 cdot 1end{array}]The gcd in terms of 4 and 9 is written as ( 1=9-2 cdot 4 ).

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Prove the following statement by the use of induction. "An arrangement of n>2 lines in a general position (no 2 lines parallel, no 3 lines concurrent) partitions the plane into connected bounded and unbounded regions. prove that at least one of the regions must be a triangle. Include visualizations to support your proof.

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add vusual

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