# Thread: Can someone help me with my proof?

1. ## Can someone help me with my proof?

Hello,
I have been working on this proof for quite some time now can someone give me some pointers?

Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.

I know that if somehow I can prove that A intersected with B= empty set then I can use the laws of transitivity to come to prove my proof. I also know that if B intersected with C = empty set that means both B AND C are empty and contain no elements. If 'all the members' of A are in B and B= empty set then A= empty set as well.

Can someone help me piece this together please?

2. Suppose that $x\in A\cap C$.
That means that $x\in A$. But that means $x\in B$.
What is wrong with that?

3. I am not sure, it is hard for me to come up with an answer from that hint thanks though....

I was thinking something along the lines of "x cannot be both an element of A and an element of B because......." but then I get stuck.

4. Originally Posted by matthayzon89
Hello,
I have been working on this proof for quite some time now can someone give me some pointers?

Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.

I know that if somehow I can prove that A intersected with B= empty set then I can use the laws of transitivity to come to prove my proof. I also know that if B intersected with C = empty set that means both B AND C are empty and contain no elements. If 'all the members' of A are in B and B= empty set then A= empty set as well.

Can someone help me piece this together please?

you want to prove that :

$A\cap C =\emptyset$

Because you know that the empty set is a subset of each set ,then all you have to prove is :

$A\cap C\subset\emptyset$

OR

xεΑ and xεC => $x\in\emptyset$

5. Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.

Okay so this is my proof attempt:

Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

We know that empty set is the subset of each set. So, empty set is the subset A, empty set is the subset of B and empty set is the subset of C.

Since, empty set is the the subset of A and B then A intersect B is an empty set. If A intersect B= empty set and B intersect C equals empty set then A intersect C equals empty set (By laws of transitivity). **END OF PROOF**

P.S
Is the set A with a bar on top of it notation the same as the complement of A?

6. Originally Posted by matthayzon89
Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.

Okay so this is my proof attempt:

.

Since, empty set is the the subset of A and B then A intersect B is an empty set. If
WRONG :

Let A ={1,2} B={1,5} IS $A\cap B=\emptyset$????

Yet $\emptyset\subset A\cap B$

7. I GET THE GENERAL IDEA NOW!

Since A is the subset of B (A is IN B) and B intersect C have no elements in common (empty set, meaning they don't 'overlap'), then OF COURSE that A and C DONT have any elements in common as well (they also dont overlap because A is IN B).

8. Originally Posted by matthayzon89

Darn this problem! can someone give me another hint please? ...im as lost as can be.

Since $A\subset B$ => xεA => xεB.

You also have xεC ,SO now you have that xεB and xεC .

BUT .....can you curry on from here now??

9. Originally Posted by matthayzon89
I GET THE GENERAL IDEA NOW!

Since A is the subset of B (A is IN B) and B intersect C have no elements in common (empty set, meaning they don't 'overlap'), then OF COURSE that A and C DONT have any elements in common as well (they also dont overlap because A is IN B).
Yes ,but you have to show that

10. Originally Posted by xalk
Yes ,but you have to show that
Here it is:

Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

Let x be an element of set A. If x E A and B intersect C= empty set, then x is NOT an element of C. Which means A is NOT a subset of C. Therefore, A intersect C= empty set as well (A and C cannot intersect and result in a NON-empty set if A is not a subset of C). If A intersect C= empty set and B intersect C= empty set, then A intersect C= EMPTY SET by laws of transitivity.

11. Originally Posted by matthayzon89
Here it is:

Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

Let x be an element of set A. If x E A and B intersect C= empty set, then x is NOT an element of C.
Let:

A={1,2}, B={3,5} ,C={1,7} .Here we have 1εA , $B\cap C=\emptyset$.
Yet 1εC

Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

Let x be an element of set A. If x E A and B intersect C= empty set, then A is NOT a subset of C. Therefore, A intersect C= empty set as well (A and C cannot intersect and result in a NON-empty set if A is not a subset of C). If A intersect C= empty set and B intersect C= empty set, then A intersect C= EMPTY SET by laws of transitivity.

13. Originally Posted by matthayzon89

When you solve a problem ,think before write or say anything.

Ask yourself what is given in a problem and what is asked to prove .

In a proof you must use all the given ,otherwise some where you must have gone wrong

14. Originally Posted by xalk
In a proof you must use all the given ,otherwise some where you must have gone wrong
My friend forgot to do that in a course final exam, but apparently he only proved a stronger result, so it's not always bad :P

On topic, matthayzon89 - take xalk's advice: Let $x \in A \cap C$, then $x \in A$ and $x \in C$. Using the information you are given, what will you do next?

15. Suppose that A,B, and C are sets.
Let x E A and x E C.
A is a subset of B. If A is a subset of B and x is an element in A then x is ALSO an element in B.
If the intersect of all the elements of B AND C= empty set
we can conclude that x= empty set?

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