1. Translations in predicate logic (a) Let Parent(x,y) be a relation stating that x is a parent of y. Write a predicate logic formula stating that x is an uncle or an aunt of y. Use only predicates Parent and =. Flint: variables can take the same value, unless you explicitly tell them not to using “=”. (b) Now write a formula stating that x has a sibling, also using predicates Parent and =. 2. Domains and predicates (a) Consider a sentence Vx3y3z(13(x,y) A P(x,z) Q(y,z)). Let all quantifiers have the same domain, consisting of two elements a,b. Now, give interpretations of P and Q over this domain that make this formula true. Finally, give another pair of interpretations of P and Q that makes this formula false. (b) Suppose that the formula is Vx3y3z(P(x,y)AP(x,z) P(y,z)), that is, there is no Q. What is the value of this formula on the empty domain? On the domain consisting of only one element (for any interpretation of P)? 3. Predicate logic reasoning; Let the domain be a group of three cats, Tiger, Ashes and Smokey. Consider the following premises VxVy(Kitten(x)A-Kitten(y) Parent(y,x)) VxVy(Parent(x,y) —>-Kitten(x)) V xVy(Parent(x,y) Kitten(y)) Parent(Smokey,Ashes) 4. Proofs . Use direct proof technique to show that product of two odd numbers is odd. Use the definition of an integer n being odd if there is another integer k such that n = 2k +1