1. For the problems on this assignment, we will be working with toy models that consist of drawings of shapes. 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 0 ). To rule out trivial examples, all toy models must include at least three shapes. You can use the same shape more than once in a single model if you want, although if you do that, I recommend making something different about them (such as the size or orientation), so that it's easier to talk about them. We will use the following definitions: - T(x) means " x is a triangle." - C(x) means " x is a circle." - S(x) means " x is solid." (a) Draw a toy model that satisfies the sentence "All triangles are solid," but does not satisfy the sentence "All solid shapes are triangles." If you think no such model exists, explain why not. (b) Translate "All triangles are solid," into a formula of first-order logic. (c) Draw a toy model that satisfies the formula x(S(x)C(x)), but does not satisfy x(S(x)C(x)) ? If you think no such model exists, explain why not. (d) Draw a toy model that satisfies the formula x(S(x)C(x)), but does not satisfy x(S(x)C(x)) ? If you think no such model exists, explain why not. (e) Translate x(S(x)C(x)) into a natural sounding English sentence. (Remember that in this class, "natural sounding sentences" don't have variables in them!) (f) Which of the following toy models satisfy the formula x(S(x)T(x)) ? As always on problems like these, there may be more than one model that satisfies the formula (or none at all). You must list all the satisfying models to earn full credit. (g) Translate x(S(x)T(x)) into a natural sounding English sentence. (h) Which of the following toy models satisfy the formula x(S(x)T(x)) ? M1: Translate x(S(x)T(x)) into a natural sounding English sentence