List All Relations on \(A=\{a,b\}\) and Find Their Number
Question
Let \[ A=\{a,b\}. \]
List all relations on \(A\) and find their number.
Solution
\[ A\times A= \{ (a,a),(a,b),(b,a),(b,b) \} \]
A relation on \(A\) is any subset of \[ A\times A. \]
All relations are:
\[ \phi \]
\[ \{(a,a)\} \]
\[ \{(a,b)\} \]
\[ \{(b,a)\} \]
\[ \{(b,b)\} \]
\[ \{(a,a),(a,b)\} \]
\[ \{(a,a),(b,a)\} \]
\[ \{(a,a),(b,b)\} \]
\[ \{(a,b),(b,a)\} \]
\[ \{(a,b),(b,b)\} \]
\[ \{(b,a),(b,b)\} \]
\[ \{(a,a),(a,b),(b,a)\} \]
\[ \{(a,a),(a,b),(b,b)\} \]
\[ \{(a,a),(b,a),(b,b)\} \]
\[ \{(a,b),(b,a),(b,b)\} \]
\[ \{(a,a),(a,b),(b,a),(b,b)\} \]
Since \[ n(A\times A)=4, \]
Total number of relations \[ =2^4=16 \]
Therefore,
\[ \boxed{16} \]