Recent content by AbelAkil

Yeah...I get it. Thanks very much. In addition, how to prove part (b), that is how can I show that both H and K are partitioned by finite cosets of H \cap K... I appreciate your insightful answer!

Sorry, I made some mistakes when I wrote the post. In fact, I mean the intersection of H and K is a subgroup of both H and K...Could U give me some tips to prove it?

Homework Statement (a)Let H and K be subgroups of a group G. Prove that the intersection of xH and yK which are cosets of H and K is either empty or else is a coset of the subgroup H intersect K (b) Prove that if H and K have finite index in G then the intersection of H and K also has finite...

We should choose the end points of each segments carefully to make sure that the end points are all irrational numbers and the set is still closed!

yes, you are right...sorry, I made a mistake in my proof... but now I can understand it...thank U very much...

How to modify it? Rational numbers are dense in R...

Homework Statement Is there any nonempty perfect set in R which contains no rational number? Homework Equations A set E is perfect iff E is closed and every point of E is a limit point of E The Attempt at a Solution We should avoid rational numbers to become limit points, so we...

Is there any nonempty perfect set in R which contains no rational number? I cannot figure it out...:frown: I appreciate your solutions! PS: A set E is perfect iff E is closed and every point of E is a limit point of E

“What is the mathematics” by Richard Courant is the best I think!