Abstract Algebra Answers

Let S be the set of all real numbers except −1. Define * on S by

a * b = a + b + ab

Show that〈 S,*〉is a group and solve 2 * x * 3 = 7 in S

.

Prove that a group has exactly one element with the property that

x * x = x (this is called an idempotent element)

Prove that if phi : S ---> T is an isomorphism of <S,*> with <T, #> and psi: T ---> U is an isomorphism of <T, #> with <U, diamond> then the composite function psi composed with phi is an isomorphism of <S, *> with <U, diamond>

Find a binary function * on Q such that phi is an isomorphism mapping <Q, +> with <Q , *) and give the identity element for *. The map is phi : Q --> Q is given by phi(x)=3x-1 for x is an element of Q

Give a proof that the associative property is a structural property. (Start with two isomorphic structures, let the operation on one be associative and show that its image must also be associative).

Determine whether or not the given map is an isomorphism on the structures . If it isn’t explain why. Let F be the set of all functions f mapping R —> R that have derivatives of all orders . <F,+> with <R,+) with phi (f) = f’(0) for f is an element of F

Suppose * is an associative and commutative binary operation of a set S. Show that H = { a is an element of S | a*a=a} is closed under the operation *. 

You are a farmer about to harvest the crop. To describe the uncertainty in the size of the harvest, you feel that it may be described as normal distribution with a mean of 80,000 bushels with a standard deviation  of 2500 bushels. Find the probability that your harvest will exceed 84,400 bushels.

Give a proof that the associative property is a structural property. (Start with two isomorphic structures, let the operation on one be associative and show that its image must also be associative).

Prove or give a counter example: “Every commutative binary operation on a set having just 2 elements is associative.”