[10] 1. Prove that that the natural numbers with the binary operation of multiplication (as defined in the video) forms a commutative monoid. Furthermore, prove that multiplication distributes over addition. Hint: First, you need to use induction to prove that given function f: X →X, (f")" = (f")" [10] 2. Use induction to prove 1+4+9+...+n2 = n(n+1)(2n+1) 6.

Q: Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0,0)…

A:

Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

A:

Q: 2p 4-1(e) + Joi(e) = 空p(e). %3D

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: a) If q(n) = byn* + Bk-1n*-1+...+ b1n+ bo is a polynomial in the variable ne N with be # 0, show…

A:

Q: a) Factor a* + a3 +x into irreducible polynomials in Za[z]; b) Find the multiplicative inverse of…

A: The multiplicative inverse of (x2+1) in given ring is square (x2-1). For justification locate step 2…

Q: Find an algebraic integer a in a Trace(a) = 17. quadratic field with N(a) 31 and

A: To find: An algebraic integer α in a quadratic field with Nα=31 and Traceα=17.

Q: ould we also use Cantor's diagonalization to contradict this as well?

A: In the question, we have to check that the set of all functions f: N → {0, 1} is countable or…

Q: Provide a combinatorial proof (argue that both sides count the same thing) for the following:…

A: Please see the below picture for detailed solution.

Q: Determine whether e(x)=15x+4(mod 26) and e(x)= 4x+5(mod 26) are invertibles or not. If yes, find the…

A: We are asked to determine if the given equation are invertible and if they are we have to find the…

Q: Prove that if f(n) is multiplicative, then so is f(n)/n.

A: We have to prove that if f(n) is multiplicative, then f(n)n is also a multiplicative function.

Q: Consider the function f.N×N¬N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2)…

A: Given:- (1) Base case: Let m∈ℕ and define f(0,m)=0......(1)              (2) Recursive case: For any…

Q: 1. Let J₁ = {0, 1,2,3} and define f: J₁ J₁ and g: J₁ J₁ by f(x) = (x + 3)² mod 4 and g(x)=x²-3x +- 5…

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.