Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
cinar
Group Title
\[ Let f:A \rightarrow B $ be a given function. Prove that f is onetoone (injective) $ \Leftrightarrow f(C\cap D)=f(C)\cap f(D) $ for every pair of sets C and D in A $\]
 one year ago
 one year ago
cinar Group Title
\[ Let f:A \rightarrow B $ be a given function. Prove that f is onetoone (injective) $ \Leftrightarrow f(C\cap D)=f(C)\cap f(D) $ for every pair of sets C and D in A $\]
 one year ago
 one year ago

This Question is Closed

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
Let \[f:A\rightarrow B\] be a given function. Prove that f is onetoone (injective) \[\leftrightarrow f(C\cap D)=f(C)\cap f(D)\] for every pair of sets C and D in A
 one year ago

cinar Group TitleBest ResponseYou've already chosen the best response.0
\[Let f:A\rightarrow B be a given function. Prove that f is onetoone (injective) \\\Leftrightarrow f(C\cap D)=f(C)\cap f(D) for every pair of sets C and D in A\]
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
i was just rewriting so i could read it, i am not sure i know how to do it
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
well one way is trivial, since \(f(A\cap B)\subset f(A)\cap f(B)\) for any \(f\)
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
or does that need clarification as well? we can write it out if you like
 one year ago

cinar Group TitleBest ResponseYou've already chosen the best response.0
yes we need to right it..
 one year ago

cinar Group TitleBest ResponseYou've already chosen the best response.0
why letter is not separated
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
suppose \(z\in f(A\cap B)\) then \(z=f(x)\) for some \(x\in A\cap B\) making \(x\in A\) and \(x\in B\) so \(z\in f(A)\) and \(z\in f(B)\) therefore \(z\in f(A)\cap f(B)\)
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
this shows for any \(f\) you have \(f(A\cap B)\subset f(A)\cap f(B)\)
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
now we need to prove that if \(f\) in injective, we have \(f(A\cap B)=f(A)\cap f(B)\) since we already have containment one way, this amounts to showing \[f(A)\cap f(B)\subset f(A\cap B)\]
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
pick a \(z\in f(A)\cap f(B)\) so there exists a \(x_1\) in \(A\) with \(f(x_1)=z\) and likewise there is a \(x_2\) in \( B\) with \(z=x_2\) now comes the "injective" part since \(f\) is injective, and \(f(x_1)=f(x_2)=z\) we know \(x_1=x_2\) and so \[z\in f(A\cap B)\]
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
typo there, i meant "likewise there exists \(x_2\in B\) with \(f(x_2)=z\) sorry
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
so that is the proof one way, that "if \(f\) is injective, then \(f(A\cap B)=f(A)\cap f(B)\)
 one year ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
other way is easier, since a singleton is a set
 one year ago

cinar Group TitleBest ResponseYou've already chosen the best response.0
A={x} B={y} you mean like this one..
 one year ago
See more questions >>>
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.