For this problem, we will be making little toy models fitting certain requirements. For each part, you will draw a few small shapes to serve as a universe. Some of the shapes will be triangles (like or ), and some of them will be other shapes (like or ). Some of the shapes will be outlined (like , , or ), and others will be solid (like , , or ). For each part, you should draw a separate picture, meeting the requirements for that part. To rule out trivial examples, you must include at least three shapes in your picture. You can use the same shape more than once in each problem if you want.

Use the following definitions. T(x): x is a triangle. S(x): x is solid. Remember that a model satisfies a formula if the formula is true under that model.

If you think that the request is impossible, you must explain why.

(a) Draw a toy model satisfying xS(x), but not xS(x).

(b) Draw a toy model satisfying both xT(x) and xS(x).

(c) Draw a toy model that does not satisfy xT(x).

(d) Draw a toy model satisfying xT (x) S(x). Just to keep things from being too easy, there must be at least two triangles, at least two squares, at least two outlined shapes, and at least two solid shapes. (You can use the same shape to fit more than one of these requirements, for example a solid square counts as one of your squares and as one of your solid shapes.)