hw1sol_Math113_Fall2008

# hw1sol_Math113_Fall2008 - Homework 1 selected solutions...

• Notes
• 1

This preview shows page 1. Sign up to view the full content.

Homework 1, selected solutions Math 113: Introduction to Abstract Algebra (Sections 2, 4) 0.18 . We define an injective (one-to-one) map ϕ of B A onto P ( A ). Let f B A and let ϕ ( f ) := { x A | f ( x ) = 1 } . Suppose that ϕ ( f ) = ϕ ( g ). Then f ( x ) = 1 if and only if g ( x ) = 1. Because the only possible values for f ( x ) and g ( x ) are 0 and 1, we see that f ( x ) = 0 if and only if g ( x ) = 0. Consequently f ( x ) = g ( x ) for all x A so f = g and ϕ is injective. To show that ϕ is surjective (onto P ( A )), let S A , and let h : A → { 0 , 1 } be defined by h ( x ) = 1 if x S and h ( x ) = 0 otherwise. Clearly ϕ ( h ) = S , showing that ψ is indeed surjective.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4.29 . Let S := { x ∈ G | x-1 6 = x } . Then S has an even number of elements because its elements can be grouped in pairs x, x-1 . Because G has an even number of elements, the number of elements in G but not in S (the set G \ S ) must be even. The set G \ S is nonempty because it contains e . Thus there is at least one element of G \ S other than e , that is, at least one element other than e that is its own inverse....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern