$Let \ $ $A=\left \{ 0,1 \right \}\\ A \times A=\left \{ (0,0),(0,1),(1,0),(1,1) \right \}$
$A \ relation \ on \ a \ set \ A \ is \ the \ subset \ of \ A\times A.$
$Number \ of \ relation=2 \times 2 \times 2 \times 2=16$
$R_{1}=\left \{ \right \}\\ R_{2}=\left \{ (0,0) \right \}\\ R_{3}=\left \{ (0,1) \right \}\\ R_{4}=\left \{ (1,0) \right \}\\ R_{5}=\left \{ (1,1) \right \}\\ R_{6}=\left \{ (0,0),(0,1) \right \}\\ R_{7}=\left \{ (0,0) ,(1,0)\right \}\\ R_{8}=\left \{ (0,0),(1,1) \right \}\\$
$R_{9}=\left \{ (0,1),(1,0) \right \}\\ R_{10}=\left \{ (0,1),(1,1) \right \}\\ R_{11}=\left \{ (1,0),(1,1)\right \}\\ R_{12}=\left \{ (0,0),(0,1),(1,0)\right \}\\ R_{13}=\left \{ (0,0),(0,1),(1,1)\right \}\\ R_{14}=\left \{ (0,0),(1,0),(1,1)\right \}\\ R_{15}=\left \{ (0,1),(1,0),(1,1)\right \}\\ R_{16}=\left \{ (0,0),(0,1),(1,0),(1,1)\right \}\\$
$Number \ of \ relation \ contains \ the \ pair \ (0,1)=8$
$Alternate \ method :- $
$Let \ $ $A=\left \{ 0,1 \right \}\\ A \times A=\left \{ (0,0),(0,1),(1,0),(0,0) \right \}$
$A \ relation \ on \ a \ set \ A \ is \ the \ subset \ of \ A\times A.\\$
$Total \ number \ of \ relation=2 \times 2 \times 2 \times 2=16 \\$
$Relation \ which \ does \ not \ include \ pair \ (0,1)=2 \times 2 \times 2=8$
$Number \ of \ relation \ contains \ the \ pair \ (0,1) = \ Total \ relation \ - \ Relation \ which \ does \ not \ contains \ pair \ (0,1)$
$ = 16-8$
$=8$