let $A\;, B$ and $C$ be subsets of $E$. show that
$$A\cup B =E \iff \overline A\subset B$$
Notation :
- Absolute complement of $A$: $\overline A=\mathcal{C}_{E}^{A}=\{ x\in E \mid x\notin A \}$
My thoughts
- Method $1$ : Via Logic theory
$$A\cup B =E \iff \overline A\subset B$$ we can see that as Tautology in logic theory and we can prove it by truth table: $$(A\vee B)\iff ((\neg A)\to B)$$
\begin{array} {|c|} \hline A & B & (A\vee B )& \neg A & (\neg A \to B )& [(A\vee B)\iff ((\neg A)\to B) ]\\ \hline T & T & T & F & T & T \\ \hline T & F & T & F & T & T\\ \hline F & T & T & T & T & T\\ \hline F & F & F & T & F & T\\ \hline \end{array}
- Method $2$ : Via Venn diagram ( Venn diagram not rigorous so they are not really proofs )
I think I have two possibilities to draw Venn diagram for $A\cup B =E \implies \overline A\subset B$
$1$. $B=\overline A\qquad $ ($A\cup B=A\cup \overline A =E \implies \overline A\subset B$ )
$2$. $B=E\qquad $ ($A\cup B=A\cup E =E \implies \overline A\subset B$ )
Method $3$
- First step : show that : $\overline A\subset B \implies A\cup B =E$
Suppose that $\overline A\subset B$ and let's prove that $ A\cup B =E$
- we have $A \subset E$ and $B \subset E$ then $$(A\cup B) \subset E\quad (1)$$
- Conversely, let $x\in E $ we have two cases $x\in B $ or $x\notin B$
First case: if $x\in B$ then $x\in A\cup B $
Second case: if $x\notin B$ then $x\notin \overline A\quad (\text{since}\; \overline A \subset B )$ thus $x\in A $
Therefore $x\in A\cup B $ so in both cases we have : $$x\in E\implies x\in A\cup B$$ Therefore $$E\subset (A\cup B)\quad (2) $$ From $(1)$ and $(2)$ we conclude that $$E=A\cup B$$
- Second step : show that : $A\cup B =E \implies \overline A\subset B$
Suppose that $A\cup B=E$ and and let's prove that $\overline A\subset B$ Let $x\in \overline A$ we have: \begin{align*} x\in \overline A &\implies x\in E \\ &\implies x\in A\cup B \\ &\implies x\in B (\;\text{since}\; x\notin A )\\ x\in \overline A &\implies x\in B \end{align*} Therefore $$A\cup B =E \implies \overline A\subset B$$
Conclusion:
$$A\cup B =E \iff \overline A\subset B$$
- Is my proof correct also I'm interested in more ways of proving it