Problem 5 (10 points)

In this problem, the domain is the set of all faces of a truncated icosahedron (also known as a soccer ball). If you dont known what that looks like, just search the web for an image of a soccer ball. (Usually the hexagons are white and the pentagons are black.) Consider the following predicates:

P(x) = x is a pentagon

H(x) = x is an hexagon

B(x,y) = x borders y, i.e., faces x and y share an edge. (For the purpose of this exercise, a face is not considered to border itself, i.e., B(x, x) is always false.) Consider the following statements:

No two pentagons border each other

Every pentagon borders some hexagon

Every hexagon borders another hexagon

Every two hexagons border each other

Every two pentagon border each other

There is a pentagon that borders only hexagons. For each of the above statements, do the following:

(a) Indicate which of the above statements are true. (b) Write the statements in predicate logic.

(c) Negate the statements from (b). Simplify the negated statements so that no quantifier or connective lies within the scope of a negation.

(d) Translate the negated statements back into English.