PLEASE ANSWER CLEARLY AND CORRECTLY ALL THE QUESTIONS BELOW 1. Which type of pro

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PLEASE ANSWER CLEARLY AND CORRECTLY ALL THE QUESTIONS BELOW

1. Which type of proof is being used in each case? (UODpositive integers) Prove: If n is odd then (n+1 )2 is even (a) 1. n is odd 2. (n+1) is odd 3. some inference... (b) 1. (n+1) is odd 2. some inference.... (c) 1. n is odd 2. some inference.. 2. Write a proof by contradiction of this proposition: If m and n are perfect squares, then m'n is a perfect square 3. Write a contrapositive proof of this proposition: If m /n is a positive integer, then mn 4. For an inductive proof of the proposition P(n): 1*1!+ 2"2! +... + (n-1)(n-1) n!-1 for all positive integers, n > 1 (a) Write the basis step (hint:start at 2, since n > 1) (b) Write the Inductive Hypothesis (c) Complete the proof 5. For an inductive proof of the P(n): nn n! for all positive integers, n (a) Write the basis step (b) Write the Inductive Hypothesis (c) Complete the proof

Explanation / Answer

Hi,this question has multiple questions, it is against chegg policy, and we are not allowed to more than one, please understand and post others as separate questions
1. given claim is if n is odd then (n+1)2 is even,
a.we are giving an example where n is odd but (n+1)2 is odd, hence this is proof by counter example, we are giving an example to disprove the claim
b.here we are assuming (n+1)2 is odd, and proving or disproving, this is proof by contradiction, where in we assume the reverse and then prove our assumption is wrong.
c. n is odd, we are taking the given condition only and proving, hence this is direct proof.
2.Given m,n are perfect squares then mn is perfect square,
lets assume mn is not a perfect square
now since m,n are perfect squares, we have m=x*x and n=y*y for some x,y in N
then mn= x*x * y*y = xy * xy which is a perfect square, which is contradicing from our original assumption,
hence the claim is true

Thumbs up if this was helpful, otherwise let me know in comments

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