I don't understand what they mean by on set $A$ and the meaning of binary expansion of same length does the question mean cartesian product?

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    $\begingroup$ A relation on a set $A$ is simply a subset of $A\times A$. Here it is the set of ordered pairs $\langle a,b\rangle\in A\times A$ such that when you write the integers $a$ and $b$ in binary notation, they have the same number of digits (bits). For instance, $5=101_{\text{two}}$ and $6=110_{\text{two}}$, so both binary expansions have $3$ digits, and $\langle 5,6\rangle\in R$. But $3=11_{\text{two}}$ has only $2$ digits in its binary expansion, so $\langle 2,5\rangle\notin R$. $\endgroup$ – Brian M. Scott May 19 at 20:41

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