A community for students.
Here's the question you clicked on:
 0 viewing
 2 years ago
could someone explain what equinumerous means in mathematical Theoretical course?!
 2 years ago
could someone explain what equinumerous means in mathematical Theoretical course?!

This Question is Closed

manjuthottam
 2 years ago
Best ResponseYou've already chosen the best response.0the definition states " two sets S and T are called equinumerous, and we write S ~ T, if there exists a bijective function from S onto T". Bijective definition is "a function is bijective if it is surjective and injective". I'm asked to "Prove that if (S \ T) is equinumerous to (T \ S), then S is equinumerous to T" how do i do that?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.1What exactly do you mean by the S\T and T\S? Are those set minuses?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.1So there are the same number of elements in S that aren't in T as there are elements in T that aren't in S. Let \(U=S\cap T\). So \(S\setminus T=S\setminus (S\cap T)\). Then \(S=S\setminus T +U\). Now let \(f:S\setminus T\to T\setminus S\) such that \(f\) is a bijection. Now define\[g:S\setminus T+U\longrightarrow T\setminus S+U\]by\[ g(s)=\begin{cases} f(s)\qquad s\in S\setminus T \\ s\qquad \;\;\;\;\,s\in U \end{cases}\]

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.1Note that if we restrict \(g\) to only the domain \(U\), we get a bijective function since \(g(g(s))=s\). I.e., \(g\) has an inverse. Similarly, if we restrict \(g\) to only the domain \(S\setminus T\), then we have a bijective function since \(f\) is bijective. So \(g\) is bijective over its whole domain, and is therefore a bijective function. Finally, since \(S\setminus T+U=S\) and \(T\setminus S+U=T\), \(S\) and \(T\) are equinumerous.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.1Did that all make sense?

manjuthottam
 2 years ago
Best ResponseYou've already chosen the best response.0oh yes thank you!!
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.