Can A be a subset of C if it's disjoint from B?

[iHeartof12]

Let A, B and C be sets.
Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC.

My attempted solution:
Assume A\subseteqB\cupC and A\capB=∅.
Then \veex (x\inA\rightarrowx\inB\cupx\inc).

I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.

 

[jedishrfu]

...disregard earlier post...

 

[jbunniii]

Suppose x \in A. The goal is to show that this implies x \in C.

Since x \in A and A \subset B \cup C, it follows that x \in B or x \in C. Can you exclude one of these possibilities?

 

[iHeartof12]

Since A\bigcapB=∅, x\inA or x\inB.
Thus x\inA, x\notinB and x\inC.
Therefor A\subseteqC.

Is that a good way to show how to exclude the possibility of x\inB?

 

[jbunniii]

Since A\bigcapB=∅, x\inA or x\inB.
Thus x\inA, x\notinB and x\inC.
Therefor A\subseteqC.

You have the right idea, but the wording is a little unclear. The following is not true: "Since A\bigcapB=∅, x\inA or x\inB."

How about the following:

Suppose x \in A. The goal is to show that this implies x \in C.

Since x \in A and A \subset B \cup C, it follows that x \in B or x \in C. However, x cannot be in B, because if it were, then we would have x \in A \cap B = \emptyset, which is impossible. Therefore...

 

[iHeartof12]

Ok I think I get it tell me if I worded this correctly:

Suppose x \in A and A \cap B = \emptyset
Since x \in A and A \subset B \cup C, this means that x \in B or x \in C. Consequently, x cannot be in B, because if it were, then we would have x \in A \cap B = \emptyset, which is impossible. Therefore x \in C. Thus x \in A implies x \in C so it follows that A \subseteq C

 

[jbunniii]

Ok I think I get it tell me if I worded this correctly:

Suppose x \in A and A \cap B = \emptyset
Since x \in A and A \subset B \cup C, this means that x \in B or x \in C. Consequently, x cannot be in B, because if it were, then we would have x \in A \cap B = \emptyset, which is impossible. Therefore x \in C. Thus x \in A implies x \in C so it follows that A \subseteq C

Looks good to me.

 

[iHeartof12]

ok thank you for your help