Question
Which of the following is/are true ?
I. If $$A$$ is a subset of the universal set $$U,$$ then its complement $$A'$$ is also a subset of $$U$$.
II. If $$U = \left\{ {1,\,2,\,3,.....,\,10} \right\}$$ and $$A = \left\{ {1,\,3,\,5,\,7,\,9} \right\},$$ then $$\left( {A'} \right)' = A.$$
A.
Only I is true
B.
Only II is true
C.
Both I and II are true
D.
None of these
Answer :
Both I and II are true
Solution :
If $$A$$ is a subset of the universal set $$U,$$ then its complement $$A'$$ is also a subset of $$U.$$
We have, $$A' = \left\{ {2,\,4,\,6,\,8,\,10} \right\}$$
Hence, $$\left( {A'} \right)' = \left\{ {x:x\, \in \,U{\text{ and }}x\, \notin \,A'} \right\} = \left\{ {1,\,3,\,5,\,7,\,9} \right\} = A$$
It is clear from the definition of the complement that for any subset of the universal set $$U,$$ we have $$\left( {A'} \right)' = A$$