Theory:

Important!
Let us recall complement of a set.
Let \(U\) be the universal set containing finite sets \(A\) and \(B\), then:
 
(i) \((A \cup B)^{\prime}\) \(=\) \(A^{\prime} \cap B^{\prime}\)
 
(ii) \((A \cap B)^{\prime}\) \(=\) \(A^{\prime} \cup B^{\prime}\)
Example:
1. Let \(U\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\), \(A\) \(=\) \(\{10\), \(30\), \(50\), \(60\), \(70\}\) and \(B\) \(=\) \(\{10\), \(20\), \(30\), \(50\), \(80\}\).
 
Verify that \((A \cup B)^{\prime}\) \(=\) \(A^{\prime} \cap B^{\prime}\).
 
L.H.S: \((A \cup B)^{\prime}\)
 
\(A \cup B\) \(=\) \(\{10\), \(30\), \(50\), \(60\), \(70\}\) \(\cup\) \(\{10\), \(20\), \(30\), \(50\), \(80\}\)
 
\(A \cup B\) \(=\) \(\{10\), \(20\), \(30\), \(50\), \(60\), \(70\), \(80\}\)
 
\((A \cup B)^{\prime}\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\) \(-\) \(\{10\), \(20\), \(30\), \(50\), \(60\), \(70\), \(80\}\)
 
\((A \cup B)^{\prime}\) \(=\) \(\{40\), \(90\}\) - - - - - - (I)
 
R.H.S: \(A^{\prime} \cap B^{\prime}\)
 
\(A^\prime\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\) \(-\) \(\{10\), \(30\), \(50\), \(60\), \(70\}\)
 
\(A^\prime\) \(=\) \(\{20\), \(40\), \(80\), \(90\}\)
 
\(B^\prime\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\) \(-\) \(\{10\), \(20\), \(30\), \(50\), \(80\}\)
 
\(B^\prime\) \(=\) \(\{40\), \(60\), \(70\), \(90\}\)
 
\(A^{\prime} \cap B^{\prime}\) \(=\) \(\{20\), \(40\), \(80\), \(90\}\) \(\cap\) \(\{40\), \(60\), \(70\), \(90\}\)
 
\(A^{\prime} \cap B^{\prime}\) \(=\) \(\{40\), \(90\}\) - - - - - - (II)
 
From (I) and (II), we see that:
 
\((A \cup B)^{\prime}\) \(=\) \(A^{\prime} \cap B^{\prime}\).
 
Hence verified.

 
2. Let \(U\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\), \(A\) \(=\) \(\{10\), \(30\), \(50\), \(60\), \(70\}\) and \(B\) \(=\) \(\{10\), \(20\), \(30\), \(50\), \(80\}\).
 
Verify that \((A \cap B)^{\prime}\) \(=\) \(A^{\prime} \cup B^{\prime}\).
 
L.H.S: \((A \cap B)^{\prime}\)
 
\(A \cap B\) \(=\) \(\{10\), \(30\), \(50\), \(60\), \(70\}\) \(\cap\) \(\{10\), \(20\), \(30\), \(50\), \(80\}\)
 
\(A \cap B\) \(=\) \(\{10\), \(30\), \(50\}\)
 
\((A \cap B)^{\prime}\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\) \(-\) \(\{10\), \(30\), \(50\}\)
 
\((A \cap B)^{\prime}\) \(=\) \(\{20\), \(40\), \(60\), \(70\), \(80\), \(90\}\) - - - - - - (I)
 
R.H.S: \(A^{\prime} \cup B^{\prime}\)
 
\(A^\prime\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\) \(-\) \(\{10\), \(30\), \(50\), \(60\), \(70\}\)
 
\(A^\prime\) \(=\) \(\{20\), \(40\), \(80\), \(90\}\)
 
\(B^\prime\) \(=\) \(\{10\), \(20\), \(30\), \(40\), \(50\), \(60\), \(70\), \(80\), \(90\}\) \(-\) \(\{10\), \(20\), \(30\), \(50\), \(80\}\)
 
\(B^\prime\) \(=\) \(\{40\), \(60\), \(70\), \(90\}\)
 
\(A^{\prime} \cup B^{\prime}\) \(=\) \(\{20\), \(40\), \(80\), \(90\}\) \(\cup\) \(\{40\), \(60\), \(70\), \(90\}\)
 
\(A^{\prime} \cup B^{\prime}\) \(=\) \(\{20\), \(40\), \(60\), \(70\), \(80\), \(90\}\) - - - - - - (II)
 
From (I) and (II), we see that:
 
\((A \cap B)^{\prime}\) \(=\) \(A^{\prime} \cup B^{\prime}\).
 
Hence verified.
Important!
L.H.S. – Left Hand Side
 
R.H.S. – Right Hand Side