--%>
Assignment Help - homework help

Theorem-G satis es the right and left cancellation laws

Let G be a group.

(i) G satis es the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b.

(ii) If g ≡ G, then (g-1)-1 = g.

E

Expert

Verified

Proof:

(i) From ax = bx, we have axx-1 = bxx-1, then ae = be, then a = b. Similarly for the other case.

(ii) Temporarily denote the inverse of g-1 by h (instead of (g-1)-1). Then the defining property of h, from the axiom for inverses applied to g-1, is that

g-1h = hg-1 = e:

But g itself satis es these equations in place of h, because the axiom for inverses applied to g says that

gg-1 = g-1g = e:

Hence, since inverses are unique, h = (g-1)-1 = g, as required.

   Related Questions in Mathematics

  • Q : Problem on budgeted cash collections

    XYZ Company collects 20% of a month's sales in the month of sale, 70% in the month following sale, and 5% in the second month following sale. The remainder is not collectible. Budgeted sales for the subsequent four months are:     

    		•
  • Q : Global And Regional Economic Development

    The Pharmatec Group, a supplier of pharmaceutical equipment, systems and services, has its head office in London and primary production facilities in the US. The company also has a successful subsidiary in South Africa, which was established in 1990. Pharmatec South A

    		•
  • Q : Problem on Linear equations Anny, Betti

    Anny, Betti and Karol went to their local produce store to bpought some fruit. Anny bought 1 pound of apples and 2 pounds of bananas and paid $2.11.  Betti bought 2 pounds of apples and 1 pound of grapes and paid $4.06.  Karol bought 1 pound of bananas and 2

    		•
  • Q : State Prime number theorem Prime number

    Prime number theorem: A big deal is known about the distribution of prime numbers and of the prime factors of a typical number. Most of the mathematics, although, is deep: while the results are often not too hard to state, the proofs are often diffic

    		•
  • Q : Formal Logic It's a problem set, they

    It's a problem set, they are attached. it's related to Sider's book which is "Logic to philosophy" I attached the book too. I need it on feb22 but feb23 still work

    		•
  • Q : Numerical solution of PDE this

    this assignment contains two parts theoretical and coding the code has to be a new. old code and modified code will appear in the university website .

    		•
  • Q : Mean and standard deviation of the data

    Below is the amount of rainfall (in cm) every month for the last 3 years in a particular location: 130 172 142 150 144 117 165 182 104 120 190 99 170 205 110 80 196 127 120 175

    		•
  • Q : Problem on inverse demand curves In

    In differentiated-goods duopoly business, with inverse demand curves: P1 = 10 – 5Q1 – 2Q2P2 = 10 – 5Q2 – 2Q1 and per unit costs for each and every firm equal to 1.<

    		•
  • Q : Law of iterated expectations for

     Prove the law of iterated expectations for continuous random variables. 2. Prove that the bounds in Chebyshev's theorem cannot be improved upon. I.e., provide a distribution that satisfies the bounds exactly for k ≥1, show that it satisfies the bounds exactly, and draw its PDF. T

    		•
  • Q : Problem on reduced row-echelon The

    The augmented matrix from a system of linear equations has the following reduced row-echelon form. 280_row echelon method.jpg

    		•