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^m)^n = (f^n)^m

Q: Show that quaternion multiplication is not commutative. (That is, give an example to show that…

A:

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

A:

Q: Which of the following are continuous and piecewise polynomials with respect to the nodes o, x1, x2,…

A: Continuous and pocewise continuous of the polynomial

Q: 19. Is Q[x]/(x² – 6x + 6) a field? Why?

A:

Q: Chapter 3: 3.2, Define multiplication and addition tables so that F5 is a field and check all the…

A: Given: F5=0, 1, 2, 3, 4

Q: Find a generating function for the number of ways to make r cents' change in pennies, nickels,…

A: 1 pennies = 1 cent 1 nickel = 5 cent 1 dimes = 10 cents 1 quarter = 25 cents So the equation is, x1+…

Q: Prove that {2, 3, 4, ...} is countably infinite. You must prove that your function is bijective

A:

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: Prove that if p is a prime in an integral domain D, then p is an irreducible.

A: This question is related to abstract algebra (Ring theory)

Q: Prove that any inverse of a bijection is a bijection. Show that the inverse of a function is unique:…

A:

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: [10] 1. Prove that that the natural numbers with the binary operation of multiplication (as defined…

A:

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: State 2 Cantor Theorem

A: From any non-empty set A to its power set P(A) Surejection (onto) Can not be defined i.e. the…