Use mathematical induction to prove the statement.
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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 nBy using recursive approach perform the following first 3 problems by using functions in Java Anlyse their working mechanisms and check their correctness Comment on their T(n)Prove the following statement by contradiction.If n is an integer and n3 + 5 is odd, then n is even.
- 6. Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma 1.1.7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).Solve the following recurrences using iteration methods and Master's Theorem (if possible) a. T(n) = 2T (n/3) +3 b. T(n) = 3T (n/6) + n
- Let P(n) be the statement “the number 2^2n -1 is divisible by 3”. Use mathematicalinduction to show that P(n) is true for all n >0Use two ways to count the number of r-combinations of[n]={1,2,…,n}that contains 1 or 2 or 3. First, separate cases with Case 1 counting all r-combinations of [n] that contains 1 , Case 2 counting all r-combination of [n] that contains 2 but not 1 , and Case 3 counting allrcombination of [n] that contains 3 but not 1 or 2 . Second, count all r-combinations of [n] not containing any of1,2,3and use the subtraction rule.Prove the following statement for positive integers n is odd if and only if 3n2+4 is odd.