4. Prove that log2 n! € O(n log2 n).
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- Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod nIf a = x^(m+n)y^l, b=x^(n+l)y^m, and c = x^(l+m)y^n, Prove that a^(m-n)b^(n-1)c^(l-m) = 12. Prove from the definition that 2n2 + 100n log n + 1000 = O(n2) 3. For what values of n is 50 n lg n greater than 0.5 n2? Why do we say that 0.5 n2 is asymptotically larger, if 50 n lg n is larger for many values? (Hint: you may need to graph the functions or play around with your calculator.)