For each of the following  Prove them  formally and identify constants c and n0 if not then disprove a)f(n) = n 2 + 9n 3 2 is O(n 2 ) b)f(n) = 8n + 5 is O(n)

Q: How do we know that f(n) has an equal growth rate to g(n)

A: for that let's understand what is thita: It is defined as the tightest bound and the tightest bound…

Q: a) f(n) = 0(g(n)) → f(n) = cg(n)+ o(g(n)), for some real constant c> 0. b) f(n) = 0(g(n)) → f(n) =…

A: Theta(n) ie Θ(n) is nothing but the tighter bound within the two functions.So lets get started with…

Q: Given the following linear systems, find which one is time invariant. a. y(n) = -5x(n - 10) b. y(n)…

A:

Q: Based on the master theorem, what is the solution to T (n) = 16 (4) +n o e (n² log n) o e (n*) e…

A: - We have to get the complexity according to master's theorem.

Q: I need help with this

A: Solution -  To prove that (n+a)b=Θ(nb) for any real constants a and b, where b > 0, we need to…

Q: a. (n²+1) 10 b. √10n²+7+3 c. 2n Ig(n + 2)² + (n + 2)² lg d. 2+¹+34-1 e. [log₂ n]

A: limn→∞f(n)g(n)=x Case 1: x is constant f(n)=θ(g(n)) Case 2: x is ∞ f(n)=ω(g(n)) Case 3: x is 0…

Q: 5. Consider the following functions, f(n) and g(n): f(n) 1₁ -logn n 2k logn g(n) 1 n n k

A: The different conditions for finding asymptotic bounds are as follows: If limn→∞f(n)g(n)=0, then…

Q: Show that the following order of magnitude results hold. (a) (b) n+5 log n = O(n). 3" = O(n!).

A: We need to prove that the order of magnitude results are the same or not. See below steps for…

Q: Find the closed-form equation of T(n)=*** using substitution a) T(n)= T(n-1)+4 b)…

A: T(n) = T(n-1) + 4 = T(n-1) + (1×4)=T(n-2) + 4 + 4 = T(n-2) + (2×4)=T(n-3) + (3×4)=T(n-4) +…

Q: Use iaduction to prove that positive intege rs n?(n+1)? 3 3 こ I +2 + 3 +...t n 4

A: I m giving you the solution in the next step:

Q: Prove, by finding constants C₁, C₂, and no that satisfy the definition of order of magnitude, that…

A: The following logarithm rules were used in the this problem: log(xy) = log x + log y This rule was…

Q: Based on the master theorem, what is the solution to T (n) = 3T (2/2) + n² (n²) oe (n² logn) (n³) e…

A: Master's theorem: To apply master's theorem, the recurrence relation must be in this form: T(n) =…

Q: Prove that"1+3+5+………..+(2n-1)= n2

A: We will prove this by 2 method.

Q: Determine the growth rate of the function belo T(n) = 1+2+3+...+n T(n) = 0(n²) T(n) = O(n) T(n)= O(n…

A: The complexity of an algorithm is expressed in theta notation. It limits a function from both the…

Q: Prove the following statement by contraposition. For every integer x, if 5x 2 – 2x + 1 is even, then…

A: Proof for every integer x, if 5x2 – 2x + 1 is even, then x is odd.

Q: Let C(n) be the constant term in the expansion of (x + 7)". Prove by induction that C(n) = 7 for all…

A: Introduction Induction: A useful technique in arithmetic is induction. It is a method of…

Q: Give the bound using big-oh(O) notation on the following 2 solutions and provide a brief description…

A:

Q: If two functions f(n) and g(n) grow at the same rate asymptotically, what else can we say about…

A:

Q: 2n-1 ) Prove using induction that Σ (2j + 1) = 3n² for all positive integers j=n n.

A: :: Solution::

Q: For each of the following expressions, find if they are O(1), O(2"), O(n²), O(log n), O(n2022), O(n…

A: Below are the answers with reason:

Q: Solve the following formula to find a Big-O result for T(n) using Master Theorem. (a) T(n) = 2T…

A: Here in this question we have given some equation and we have asked to use master theorem to find…

Q: Based on the master theorem, what is the solution to T (n) = 16 (7) + n ○e (n³) e (n² log n) ○e (¹)…

A: Answer is:  Θ(n2)     The running time of divide and conquer algorithms may be calculated using the…

Q: Problem 4. Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the…

A: Here is the solution step by step -

Intelligent Machines

Machines that use artificial intelligence to learn from the environment and past experiences to perform better in the presence of variability and uncertainty are called intelligent machines. Intelligence develops according to environment and experience. It is human-like behavior. When the computer system or any programmed machine learns from the inputs and corresponding outputs, it can be called an intelligent machine.