Does proving $A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)$ prove the distributive law of sets?

I know the proof for A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C) but not the distributive property of sets but I believe it satisfies the distributive law? Is it or is it not?

  • $\begingroup$ What is the distributive law? $\endgroup$ – Jacob Wakem Apr 24 '17 at 15:24
  • $\begingroup$ @Alephnull $A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)$ $\endgroup$ – mathewconway Apr 24 '17 at 15:26
  • $\begingroup$ Actually that inequality you are referring to is valid in any lattice, and in particular is valid for sets. It is the reverse inclusion that proves the distributive law. $\endgroup$ – amrsa Apr 24 '17 at 15:48

You are halfway to proving the distributive law. You also need to show the inclusion the other way.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.