Fill in the blanks to make each of the following a true statement :

$A \cup A^{\prime}=\ldots$

Taking the set of natural numbers as the universal set, write down the complements of the following sets:

$\{ x:x$ is a prime number $\} $

Let $U=\{1,2,3,4,5,6,7,8,9\}, A=\{1,2,3,4\}, B=\{2,4,6,8\}$ and $C=\{3,4,5,6\} .$ Find

$(A \cup C)^{\prime}$

Let $U=\{1,2,3,4,5,6,7,8,9\}, A=\{1,2,3,4\}, B=\{2,4,6,8\}$ and $C=\{3,4,5,6\} .$ Find

$B^{\prime}$

If $U=\{a, b, c, d, e, f, g, h\},$ find the complements of the following sets:

$B=\{d, e, f, g\}$

If $U =\{1,2,3,4,5,6,7,8,9\}, A =\{2,4,6,8\}$ and $B =\{2,3,5,7\} .$ Verify that

$(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$