Prove that if $\prec$ is a well-ordering on a set X and if $Y \subseteq X$, then $\prec_Y=\{(x,y) \mid (x \in Y) \wedge (y \in Y) \wedge (x \prec y)\}$ is a well ordering on Y.
I am a little confused on where to start. Also I don't really know what $\prec_Y$ really means.
The relation $\prec_Y$ is the restriction of $\prec$ to $Y$. This means that $\prec_Y$ is only defined when both elements are from $Y$, and then it agrees with the original $\prec$.
Now you have to show that $(Y,\prec_Y)$ is a well-ordered set. This means linearly ordered, and if $A\subseteq Y$ is non-empty, then it has a minimal element. Note that all these properties follow from the fact that $(X,\prec)$ is a well-ordered set.