Deforming disks into other disks

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Right now I'm casually reading through Carson's "Topology of Surfaces, Knots, and Manifolds." I don't have a strong background in topology, and I was told that this was a very accessible and informative text.

The problem that I am looking at involves the four figures I have attached. I'm having problems finding way deform $X$ onto $Y$ and $Y$ onto $Z$. The problem further involves deforming $X$ and $Y$ onto $D$, but I will worry about that later. It's not a very rigorous.

I believe you can deform $X$ onto $Y$ by stretching $X$ into a rectangle. Linking two sides into a donut (making sure the interval is on the inside), and then shrinking the interval to a point.

Four shapes in quesition