Prove the statement by mathematical induction. 3+5+7+…+(2n+1)=n(n+2) 1. Proposition is true when n=1, since n(n+2)=A(1+2)=B. A= B= 2. We will assume that the proposition is true for a constant k=n, so 3+5+7+…+(2k+1)=C(k+D). C= D= 3. Then, 3+5+7+…+(2k+1)+(Ek+F)=k(k+2)+(Gk+H). E= F= G= H= 4. Thus: ∑i=1k+12k+1=k2+4k+3

Hi, the following question is about proof by contradiction, can you please help thanks w working out thanks.

Which of the following are examples of inductive reasoning (reasoning that relies on enumerative induction)? Everyone l've seen at this party is wearing a costume, therefore it's a costume party The best explanation of why things usually appear to be a particular way is that they really are that way. If you want to make ice, and water freezes at 32F, then chill the water to 32F. Everyone I've seen at this party is wearing a costume, therefore everyone here is wearing a costume. In the past water has frozen at 32F, so tomorrow it will too.

3. Proof by Contrapositive Prove the following statement by contrapositive: If 6n2 +2n+2 is odd, then n is odd.

Discrete Math

This part will explore one of the underlying mathematical ideas for a proof by induction. Assume that TCN and assume that 1 € T and that T is an inductive set. Use the definition of an inductive set to answer each of the following: (a) Is 2 € T? Explain. (b) Is 3 = T? Explain. (c) Is 4 € T? Explain. (d) Is 100 € T? Explain. (e) Do you think that T plain. = N? Ex-

Evaluate the proposition ∀∃y.P(x,y) ⇐⇒ ∃y.∀x.P(x,y) for an unknown predicate P. Could the statement be true? Must the statement be true for all P? Give a specific P (with domains for x and y) where it is either true or false. Prove that it is true for all P, or show a contradiction.

The following proof is wrong because only the converse has been proven, but what is the right way to write this proof.

Share your work would help tremendously.

How can we complete the proposition for part a) and part b)?