Is this a bound variable?

Question

If I write

$\left \{\begin{array}{llll} & y = z \\ & z = x + 2 \end{array} \right.$

could I make the argument that $z$ is a "bound" variable. I've seen it referred to as a "dummy" variable. Is it bound by the $\{$. Thanks.

Answer 1

All the variables in the two equations are free. The bracket to the left groups the equations together, meaning both equations must hold. So the expression is two equations in three variables. You probably noticed that $z$ can be eliminated, transforming the expression into the single equation $y=x+2$. In other words, it is not true that $z$ is a bound variable, but it is true that it can be easily eliminated, giving a simpler expression.

If a variable is bound, it makes no sense to substitute a number for it. In the problem, you can substitute numbers for all three variables, sometimes making the equations simultaneously true and sometimes not.