How is A ⊆ B equivalent to A ∩ !B = ∅ and !A U B = U? ! being the bar above the letter representing not because I can't figure out how to use that symbol. I know the latter two are equivalent through de Mogen's Law and !∅ = U(univeral set)

Question

How is A ⊆ B equivalent to A ∩ !B = ∅  and !A U B =  U? ! being the bar above the letter representing not because I can't figure out how to use that symbol. I know the latter two are equivalent through de Mogen's Law and !∅ = U(univeral set)

ybarrap · Answer

$$
A\subseteq B \equiv x\in A\rightarrow x\in B \equiv \bar{A}\cup B
$$
It's complement is
$$
\overline{\bar{A}\cup B}\equiv A\cap\bar{B}
$$

ybarrap · Answer

Does this make sense?

ybarrap · Answer

The 1st step is justified by the logical relation between implication and its disjunctive equivalent: http://en.wikipedia.org/wiki/Material_conditional