Each of the other five questions asks you to prove a theorem. When the theorem is stated in English, it would be an excellent idea to first rewrite it in predicate logic, and then think about the proof techniques we discussed in class, and the general "structure" of direct or indirect proofs for certain types of statements. [8] 2. Prove that the product of any four consecutive integers is one less than a perfect square [8] 3. Mr. Yanub crre, an alien computer scientist, has written an excellent algorithm whose execution requires 5n2Vn+3n -2 steps, where n is the size the input. Prove that the number of steps of the algorithm is in O(n3). Recall that f 0(g) if 3c R13n0 hIE N, n-no f(n) cg(n). 8] 4. Prove that for any three functions f, g and h from N into R+, if f E O(h) and g 0(h), then (f + gje 0(h), where f+g is the function from N into R+ defined by (1+g)(n) = f(n) + g(n). Recall that f 0(g) if 3c R+3n0 NhIE N, n-no-f(n) by using existential instantiation twice on the definition of O cg(n). Hint: start Each of the other five questions asks you to prove a theorem. When the theorem is stated in English, it would be an excellent idea to first rewrite it in predicate logic, and then think about the proof techniques we discussed in class, and the general "structure" of direct or indirect proofs for certain types of statements. [8] 2. Prove that the product of any four consecutive integers is one less than a perfect square [8] 3. Mr. Yanub crre, an alien computer scientist, has written an excellent algorithm whose execution requires 5n2Vn+3n -2 steps, where n is the size the input. Prove that the number of steps of the algorithm is in O(n3). Recall that f 0(g) if 3c R13n0 hIE N, n-no f(n) cg(n). 8] 4. Prove that for any three functions f, g and h from N into R+, if f E O(h) and g 0(h), then (f + gje 0(h), where f+g is the function from N into R+ defined by (1+g)(n) = f(n) + g(n). Recall that f 0(g) if 3c R+3n0 NhIE N, n-no-f(n) by using existential instantiation twice on the definition of O cg(n). Hint: start