-1

Prove that $$(A \cap B) \subset A \subset (A \cup B).$$

I understand $\cap$, $\subset $, $\cup$ concepts, but i wonder how to solve this problem.

El_Dorado
  • 113
  • 3
  • First of all. Suppose that $x \in A\cap B$. Then is it true that $x \in A$? Also for $x \in A$. – openspace Sep 09 '18 at 20:13
  • I'm sure you intuitively understand what $\cap, \subset, \cup$ are, but how are you with using them in proofs? – Theo Bendit Sep 09 '18 at 20:13
  • This should be standard. If $x \in A \cap B$, then by definition $x \in A$. That gives the first inclusion. The second inclusion follows that the union is defined as objects which are in either $A$ or $B$ (or nothing) so if an object is in A then it has to be in the Union – Niall Taggart Sep 09 '18 at 20:14
  • New bug on the website ? Why after editing, my name is over the OP's name and not next to it ? – idm Sep 09 '18 at 20:15
  • @idm It's fine from my perspective. Still a bug though. Maybe post here? – Theo Bendit Sep 09 '18 at 20:16
  • @TheoBendit : I have the new version of chrome on mac, you too ? (lots of problems these last days with Chrome...) – idm Sep 09 '18 at 20:18
  • @idm I'm using Firefox. – Theo Bendit Sep 09 '18 at 20:18
  • 1
    @TheoBendit: I made a post here. :) – idm Sep 09 '18 at 20:21
  • Thank you all for your comments. Theo Bendit, actually, instead, i was just considering the sets of elements of A and B, that would be all elements of belongs to A. But i didn't find a right way to prove it. – El_Dorado Sep 09 '18 at 20:24

1 Answers1

-1

$$x\in A\cap B\implies x\in A\implies x\in A\cup B.$$ what else ?

idm
  • 11,824
  • 1
    Downvoter, please explain ! How can it be downvoted since it exactly answer to the question ? – idm Sep 09 '18 at 20:22
  • 1
    I'm not the down-voter, but I do have my own criticism of this answer. What you've done is parsed the $\subset$ signs in terms of inequalities, but you haven't parsed the $\cap$ and the $\cup$. To be a full answer (not a hint), you'd need to make the connections between $\cap$ and "and", and $\cup$ and "or" more clear. The OP claims they understand the symbols, but it seems that they struggle with structuring an actual proof, so more than just a symbolic shorthand is probably called for here. – Theo Bendit Sep 09 '18 at 20:46