EXample:
If U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4,5} and B = {1,3,5},
then prove that A^{c} È B^{c} = (A Ç B)^{c}.
Solution:
A^{c} = U  A = {1,2,3,4,5,6,7,8,9}  {1,2,3,4,5}
= {6,7,8,9}
B^{c} = U  B = {1,2,3,4,5,6,7,8,9}  {1,3,5}
= {2,4,6,7,8,9}
A^{c} È B^{c} = {6,7,8,9} È {2,4,6,7,8,9}
= {2,4,6,7,8,9}
A Ç B = {1,2,3,4,5} Ç {1,3,5}
= {1,3,5}
(A Ç B)^{c} = U  (A Ç B) = {1,2,3,4,5,6,7,8,9}  {1,3,5}
= {2,4,6,7,8,9}
Therefor, A^{c} È B^{c} = (A Ç B)^{c}.
Directions: Write at least five examples of your own and prove that A^{c} È B^{c} = (A Ç B)^{c}
