Let A, B and C be sets. Prove that

  • Thread starter iHeartof12
  • Start date
*Sorry wrong section*
Let A, B and C be sets.
Prove that if A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅, then A[itex]\subseteq[/itex]C.

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

I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.
 
Last edited:
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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 [itex]\bigcap[/itex] B = ∅
 

Bacle2

Science Advisor
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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.
 

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