Solved 1 Prove The Following Statements Based On The Formal Chegg Question: 1. prove the following statements based on the formal definition. (10 points) a. 6(n∧2) 20n∈ Ω(n∧3) b. 4∧n∈o(10∧n n∧20) 2. what is the order (o) of the following complexity functions? explain your responses. (10 points) a. if n=1 then t(n)=1 if n>1 then t(n)=2t(n 2) n∗ solve this using the substitution b. I understand the math expressions, but i do not understand why we chose the ones we did, and why and how they prove anything. can you help? find the limit $$ \lim\limits {x \to 1} \ (x 4) ,$$ and prove it exists using the $\epsilon$ $\delta$ definition of limit.
Solved 1 Prove The Following Statements Based On The Formal Chegg Example 1.continued. recall the statement we wish to prove: let aand bbe integers. if ais even and adivides b, then bis also even. the structure described above indicates that we can approach this proof by assuming p(as described previously) to be true, and following a series of conclusions until we can conclude that qis also true. (d) for each integer \(a\), if 4 divides (\(a^2\) 1), then 4 divides (\(a\) 1). determine if each of the following statements is true or false. if a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. Question 1. i. 6n 2 20n ∉ Ω(n 3). to prove this, we need to show that 6n 2 20n is not asymptotically lower bounded by n 3 the formal definition of Ω(g(n)) states that there exist constants c > 0 and n 0 ≥ 1 such that f(n) ≥ c ⋅ g(n) for all n ≥ n 0 . Prove the following statements based on the formal definition. i. 6n 2 20n ∉ Ω(n3 ) the formal definition of omega (Ω) is as follows: suppose f(n) and g(n) are two functions. f(n)= Ω(g(n)) if and only if there exist constants k and c that cg(n) ≤ f(n) for all n≥k.
Solved 1 Prove The Following Statements Based On The Formal Chegg Question 1. i. 6n 2 20n ∉ Ω(n 3). to prove this, we need to show that 6n 2 20n is not asymptotically lower bounded by n 3 the formal definition of Ω(g(n)) states that there exist constants c > 0 and n 0 ≥ 1 such that f(n) ≥ c ⋅ g(n) for all n ≥ n 0 . Prove the following statements based on the formal definition. i. 6n 2 20n ∉ Ω(n3 ) the formal definition of omega (Ω) is as follows: suppose f(n) and g(n) are two functions. f(n)= Ω(g(n)) if and only if there exist constants k and c that cg(n) ≤ f(n) for all n≥k. Prove the following statements based on the formal definition. (10 points) a. 6(n∧2) 20n∈ Ω(n∧3) b. 4∧n∈o(10∧n n∧20) this problem has been solved!. Prove “the square of every odd number is odd.”. proof: let b be an arbitrary odd number. then, b = 2c 1 for some integer c (depending on b). therefore, b2 = (2c 1)2 = 4c2 4c 1 = 2(2c2 2c) 1. since 2c2 2c is an integer, b2 is odd. the statement. follows since b was arbitrary.
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