Let $X = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ be the universal set and $ A = \{0, 1, 3, 5, 6, 8\}$, and $ B = \{1, 5, 6, 8, 10, 11\}$
Find the binary representation of:
a. $ A − B $
b. $ (A ∪ B)^{c} $
c. $ (A^{c} ∩ B^{c}) $
My answers:
a. $ A − B = \{0, 3, 8\} $
Binary Representation: $ (0000, 0011, 1000) $
b. $ (A ∪ B)^{c} = \{2,4,7,9\} $
Binary Representation: $ (0010, 0100, 0111, 1001) $
c. $ (A^{c} ∩ B^{c}) = \{2,4,7,9\} $
Binary Representation: $ (0010, 0100, 0111, 1001) $
Is this correct?
$(1)$ Should be $$A-B = \{0, 3\}$$ And so the binary representation should be $(0000, 0011).$
But otherwise, everything looks good!
Note that the set in $(2)$ and the set in $(3)$ are equivalent, by DeMorgan's Rule for sets.
That is, for all sets $A, B$, $$(A \cup B)^c = A^c \cap B^c$$.