Let A, B and C be sets. Prove that

[iHeartof12]

*Sorry wrong section*
Let A, B and C be sets.
Prove that if A$\subseteq$B$\cup$C and A$\cap$B=∅, then A$\subseteq$C.

My attempted solution:
Assume A$\subseteq$B$\cup$C and A$\cap$B=∅.
Then $\vee$x (x$\in$A$\rightarrow$x$\in$B$\cup$x$\in$c).

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

 

[jedishrfu]

you almost have it.

for all x in A and A is a subset of B U C then x is in B U C
if x is in B U C then x is in B or x is in C

now just describe x with respect to the A $\bigcap$ B = ∅

 

[Bacle2]

Another perspective: you can maybe rigorize it by saying:

a in B or a in C , but a not in B , so we must have a in C:

Basically, as you rightly concluded, a must be in one

of B or C, but ,by assumption/construction, a is not in B,

so a must be in C.