Decide $\overline{A} $ U $\overline{B}$

Question

$U = \left \{ 1,2,3,4,5,6,7,8 \right \}$

$ A = \left \{ 1,2,3 \right \}$

$ B = \left \{ 3,4,5,6 \right \} $

a) Decide $ \overline{A} $ U $\overline{B} $

Correct answer:

\begin{Bmatrix} 1,2,4,5,6,7,8 \end{Bmatrix}

I don't understand that answer. What does $\overline{A} $ and $\overline{B}$ mean?

Answer 1

$\overline{A}$ means that all the elements in $U$ that are not in set A. So, $\overline{A} =\{4,5,6,7,8\}$

$\overline{B}$ means that all the elements in $U$ that are not in set B. So, $\overline{B}=\{1,2,7,8\}$

Now $\overline{A}\cup\overline{B}=\{1,2,4,5,6,7,8\}$

Answer 2

$\bar{A}$ means everything in $U$ that’s not in $A$. So we get $\bar{A} = \{ 4,5,6,7,8 \}, \bar{B} = \{ 1,2,7,8 \}$ from which the result follows.

Answer 3

You can also use $\overline{A}\cup\overline{B}=\overline{A\cap B}=\overline{\{3\}}=U\setminus\{3\}=\{1,2,4,5,6,7,8\}$

• When the context in unambiguous $\overline{X}$ designates $U\setminus X$.

• When the context in ambiguous (i.e when $\overline{X}$ designates the closure of $X$ and $\mathring X$ its interior) then $X^\complement$ is prefered.