**0. Memorize the four Rules of Inference for Proof by Induction**given in the table of the same name in lecture notes in section 7.2 (below the Theorem 19 about equivalent forms). We will have a quiz on those definitions.**1. Prove the following theorems.**Use semiformal proofs using any of the shortcuts we discussed, but for reasons you can only use the rules from Logic, the Peano Axioms, the definition of 1,2,3,… that I wrote on the board, and any theorems or lemmas we proved. You can use the results of an earlier assigned or proven problem in the proof of a later problem but not vice-versa. You cannot use “by arithmetic” as a reason. All quantiied variables are assumed to be natural numbers. Hint: You can use the Lemma we proved in class! It’s very helfpul!**(a) Theorem (addition is commutative):**$$\forall n,\forall m, m+n=n+m$$**(b) Theorem (harder than $1+1$):**$$2+2=2\cdot 2$$**(c) Theorem (associativity of addition):**$$\forall m,\forall n,\forall p, m+(n+p)=(m+n)+p$$**(d) Theorem (multiplicative identity):**$\forall n, n\cdot 1=n$ and $1\cdot n=n$(Hint: you don’t need induction to prove the first equality but you need it for the second.)**(e) Theorem (good thing we have $1+1$):**$$1\leq 2$$**(f) Theorem (just as we expected):**$\forall n,\sigma(n)=n+1$**(g) Theorem (strong induction):**Prove that part (a) implies part (c) in Theorem 19 of the Lecture Notes.