N bracerightbig D A g N is a bijective function Apply the Glueing Lemma Hence A

# N bracerightbig d a g n is a bijective function apply

• 2

This preview shows page 1 - 2 out of 2 pages.

( N ) bracerightbig D A \ g ( N ) is a bijective function. (Apply the Glueing Lemma.) Hence A A B . 2. (a) Let A,B,C be sets. Suppose A C and ( A negationslash = or B negationslash = ). Suppose B = . Then A negationslash = . Since A C , we have C negationslash = . Then Map ( A,B ) = Map ( C,B ) = . Hence Map ( A,B ) lessorsimilar Map ( C,B ). Suppose B negationslash = . Pick some z B . Define ψ Map ( C \ A,B, Ψ) by Ψ = ( C \ A ) × { z } . For any ϕ Map ( A,B ), there exists some Φ P ( A × B ) such that ϕ = ( A,B, Φ). According to the Glueing Lemma, ( C,B, Φ Ψ) is a function from C to A . Let F be the subset of Map ( A,B ) × Map ( C,B ) given by F = { (( A,B, Φ) , ( C,B, Φ Ψ)) | ( A,B, Φ) Map ( A,B ) } . The ordered triple f = ( Map ( A,B ) , Map ( C,B ) ,F ) is an injective function. 1 Hence Map ( A,B ) lessorsimilar Map ( C,B ). (b) Let A,B,C be sets. Suppose A lessorsimilar C and ( A negationslash = or B negationslash = ). Since A lessorsimilar C , we may pick some injective function g : A −→ C . We have g ( A ) C and A g ( A ). We also have g ( A ) negationslash = or B negationslash = . Therefore Map ( A,B ) Map ( g ( A ) ,B ) lessorsimilar Map ( C,B ). (c) Let A,B,D be sets. Suppose B D . Consider the inclusion function ι B of B into D . Define the function h : Map ( A,B ) −→ Map ( A,D ) by h ( ϕ ) = ι B ϕ for any ϕ Map ( A,B ). h is an injective function. (Why?) Hence Map ( A,B ) lessorsimilar Map ( A,D ).

#### You've reached the end of your free preview.

• Spring '20
• Countable set, Basic concepts in set theory, Cardinal number

### 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