anonymous
  • anonymous
how do I prove if A is a subset of (b intersect C) then A Is a subset of b and a is a subset of c.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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DebbieG
  • DebbieG
Are you sure you have stated it correctly? I'm not sure I believe it. Suppose: A={1, 2, 3} B={3, 4, 5} C={0,3,7} Then A(int)B={3} is a subset of C... but neither A nor B are subsets of C.
DebbieG
  • DebbieG
And by the way, welcome to Open Study! :) :)
goformit100
  • goformit100
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DebbieG
  • DebbieG
Maybe you are only supposed to prove it if it's true, or demonstrate that it isn't if it's false?
anonymous
  • anonymous
it says either prove or show a counterexample so it could be false
DebbieG
  • DebbieG
Haha... ok..... that makes more sense. :) Well, in that case I guess I gave you your counterexample, but I'd encourage you to come up with your own! :)
AkashdeepDeb
  • AkashdeepDeb
Hahahaha!!! :D LOL
asnaseer
  • asnaseer
I can show it using venn diagrams but am not sure how to prove it otherwise. |dw:1378062036383:dw|
anonymous
  • anonymous
yea I did that too I just need help with the actual proof
asnaseer
  • asnaseer
I haven't studied formal proofs in set theory so sorry but I cannot do more than venn diagrams. :(
DebbieG
  • DebbieG
It's not true, I demonstrated that above. Thus, you don't need a proof (as there isn't one possible, lol). You just need to show a counterexample.
asnaseer
  • asnaseer
@DebbieG - are you sure?
DebbieG
  • DebbieG
Wait a sec.... @brookekm194 did you change the problem?? that's not what was there before.
DebbieG
  • DebbieG
Sorry @asnaseer , I didn't re-read the problem... I got the notification because I had responded above, but I think the problem was stated differently.
anonymous
  • anonymous
yea
DebbieG
  • DebbieG
Before it said Show that if A(int)B is a subset of C then either A is a subset of C or B is a subset of C.
asnaseer
  • asnaseer
@brookekm194 - naughty naughty :) - It might have been prudent to indicate that you had changed the question.
DebbieG
  • DebbieG
OK, please don't do that - just close the question and then post your new one. It will be very confusing for anyone who might come to this thread because they have a similar problem. :)
asnaseer
  • asnaseer
otherwise it can get very confusing for those trying to help you out.
DebbieG
  • DebbieG
But for the new question: how do I prove if A is a subset of (b intersect C) then A Is a subset of b and a is a subset of c. Think about what it means that "A is a subset of (b intersect C)" Everything in B(int)C is in BOTH B and C, right? So if A is a subset of THAT, what does that say about A?
asnaseer
  • asnaseer
@DebbieG - I would be interested in knowing how to do formal proofs of sets as well - so I await eagerly. :)
DebbieG
  • DebbieG
Well, here you are really just using definitions about intersections and subsets, right? So assume that A is a subset of (b intersect C). Then everything that is in B(int)C is in B and is in C. A is contained in B(int)C, so everything in __ is in __ and also everything in __ is in __. Therefore, A is a subset of B and A is a subset of C. QED. :) Set theory was one of my favorite classes! :)
DebbieG
  • DebbieG
(I left the blanks above for you to fill in... have to make you do some of the work! lol) ;)
asnaseer
  • asnaseer
@DebbieG -- how does this differ from just using a venn diagram to show that it must be true?
DebbieG
  • DebbieG
Hmmm... well.... I am not sure that a Venn diagram is adequate as a formal proof. It does not state the properties of the sets that REQUIRE it to be true. I could make a Venn diagram that is an example of something being true in a particular instance that is not true "in general", know what I mean? So without the rigorous proof that resorts to the properties of the question at hand, I'm not sure you can say you've "proved" something. But the Venn diagram is a great tool for "wrapping your head around" it. :)
asnaseer
  • asnaseer
ok - I was hoping there would be some way of proving it using just pure maths (i.e. no english statements as such). :)
DebbieG
  • DebbieG
For example, the original problem in this thread (which it turned out is NOT true in general, I gave a counter-example above): Show that if A(int)B is a subset of C then either A is a subset of C or B is a subset of C. I can draw that in a Venn diagram, even though it isn't a generally true statement: |dw:1378063274004:dw|
DebbieG
  • DebbieG
You can do it without much of any English if you know all the proper notations, lol! Maybe an "if" and a "then" here and there... but the rest can just be a lot of sets, subsets, contained in, intersection, etc symbols.... :)
asnaseer
  • asnaseer
ok - thanks for taking the time to explain - much appreciated. :)
DebbieG
  • DebbieG
Sure thing! :)
anonymous
  • anonymous
\[A \subset \left( B and C \right)=\left\{ x: x \in A \rightarrow X inB and C \right\}\] \[=\left\{ x:x \in A and x \in B and \in A and x \in C \right\}\] \[= \left\{ x :(x \in A \rightarrow x \in B)and \left( x \in A \rightarrow x \in C \right) \right\}\] gives A subset B and A subset C.

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