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*Statistical Inference*by Casella and Berger. My question has to do with transformations.

Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in S_x | g(x)=y\}##. Now, ##F_Y(y)=P(Y\leq y)=P(g(X)\leq y)=P(X\leq g^{-1}(y))##.

My textbook then states ##P(X\leq g^{-1}(y))=P(\{x\in S_x | g(x)\leq y\}##.

The issue I have is with this last equality. Are we defining the meaning of ##X## less than or equal to a set here, or am I missing some intuition on sets?