2. Asymptotic Notation Exercises. (20 points) Indicate whether each of the following statements is True or False then justify your answers; if true, show the necessary constants (e.g., c,n0, etc.) that satisfy the notation, and if false, argue that no such constants exist. (a) 3n2+n+10=(n2) (b) n=(n2) (c) 10lgn=(1) (d) lgn8=O(lgn) (e) 3n=O(2n) (f) 100n3=o(n3lgn) (g) 5n10=(n10) (h) 3log7n=(log2n) (i) log2(n)=O(log(n2)) (Note: log2(n)=log(n)log(n)) (j) n21=O(1) 3. Asymptotic Notation Properties. (20 points) Let f(n) and g(n) be asymptotically non-negative functions. Using the definitions of asymptotic notation: (a) (5 points) Prove that if f(n)=o(g(n)) and g(n)=o(h(n)) then f(n)=o(h(n)). (b) (5 points) Prove that if f1(n)=O(g1(n)) and f2(n)=O(g2(n)) then f1(n)+f2(n)= O(g1(n)+g2(n)) (c) (5 points) Prove that if f(n)=O(n)+O(n2)+O(n3) then f(n)=O(n3). (d) (5 points) Disprove that if f(n)=O(g(n) ) and g(n)=(h(n)) then f(n)=(h(n)). To do this, you need to provide a counterexample by showing a specific f(n),g(n),h(n) that do not satisfy the above statement