Properties of a group
How can we say that the pair (G, o) is a group. Explain the properties which proof it.
Expert
Let G be a set and suppose that o is a binary operation on G. We say that the pair (G; o) is a group if it has the following properties.
(i) The operation o is associative; that is, (g o h) o k = g o (h o k) for all g; h; k ≡ G.(ii) There exists an identity for o ; that is, there exists e ≡ G such that g o e = e o g = g for all g ≡ G.(iii) There exist inverses for o ; that is, for each g ≡ G, there exists g-1≡ G such thatg o g-1 = g-1 o g = e:There is another property implicit in this de nition which it is useful to give a name to. Instead of saying that o is a binary operation on G, we can say that the law of closure holds for o, meaning that when o acts on two elements of G the result is also in G.
Most of the groups (G; o) we study will also have the following property.
(iv) The operation o is commutative; that is, g o h = h o g for all g; h ≡ G.
A group with this property is called commutative or, more usually, abelian, after the Norwegian mathematician Niels Henrik Abel (1802{1829).
A college student invested part of a $25,000 inheritance at 7% interest and the rest at 6%. If his annual interest is $1,670 how much did he invest at 6%? If I told you the answer is $8,000, in your own words, using complete sentences, explain how you
Explain lognormal stochastic differential equation for evolution of an asset.
A leather wholesaler supplies leather to shoe companies. The manufacturing quantity requirements of leather differ depending upon the amount of leather ordered by the shoe companies to him. Due to the volatility in orders, he is unable to precisely predict what will b
Where would we be without stochastic or Ito^ calculus?
Who had find Monte Carlo and finite differences of the binomial model?
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
Using the PairOfDice class design and implement a class to play a game called Pig. In this game the user competes against the computer. On each turn the player rolls a pair of dice and adds up his or her points. Whoever reaches 100 points first, wins. If a player rolls a 1, he or she loses all point
The basic Fermat algorithm is as follows: Assume that n is an odd positive integer. Set c = [√n] (`ceiling of √n '). Then we consider in turn the numbers c2 - n; (c+1)2 - n; (c+2)2 - n..... until a perfect square is found. If th
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
I. Boolean Algebra Define an abstract Boolean Algebra, B, as follows: The three operations are: + ( x + y addition) ( x y multiplic
18,76,764
1931478 Asked
3,689
Active Tutors
1439137
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!