Is this relation symmetric? $R={(1,1),(0,0)}\ $ R={(1,1),(0,0)}
A relation is symmetric if xRy and yRx but can it be x=y?

It symmetric.
If $\color{green}0R\color{red}0$ then $\color{red}0R\color{green}0$.
And if $\color{green}1R\color{red}1$ then $\color{red}1R\color{green}1$
And as those are the only two possible cases of $\color{green}x R\color{red}y$ possible, If we ever have $\color{green}x R\color{red}y$ it will always be the case we also have $\color{red}y R\color{green}x$.

Please let me know if you hare red/green colorblind.  If so consider this explanation:
If $\color{purple}{\large{\text{ZERO}}}R\color{orange}{\small{\text{zero}}}$ then $\color{orange}{\small{\text{zero}}}R\color{purple}{\large{\text{ZERO}}}$.
And if $\color{purple}{\large{\text{ONE}}}R\color{orange}{\small{\text{one}}}$ then $\color{orange}{\small{\text{one}}}R\color{purple}{\large{\text{ONE}}}$
And as those are the only two possible cases of $\color{purple}{\large{\text{X}}}R\color{orange}{\small{\text{y}}}$ possible, If we ever have $\color{purple}{\large{\text{X}}}R\color{orange}{\small{\text{y}}}$ it will always be the case we also have $\color{orange}{\small{\text{y}}}R\color{purple}{\large{\text{X}}}$.
