Problem 1. The purpose of this problem is to practice the use oflogic operations and quantifiers. For each Statement X belowdetermine if each of the three statementsX1, X2, X3 that follow itsatisfy the following:
a) Xi implies X;
b) X implies Xi;
c) if Xi is true then X must be false; d) if X is true then Xi mustbe false.
Statement A. In every house there is a mouse.
A1. There is no house without a mouse.A2. There exists a housewithout a mouse.A3. Mice don’t live in houses.
Statement B. For every mouse there is a blouse, such that if themouse wears the blouse he’ll get a gift from Carl FriedrichGauss.
B1. There is a mouse that can wear any blouse, but still won’tget a gift from Gauss.
B2. There are no mice for which there does not exist a specialblouse, such that if the mouse is not getting a gift from Gauss itmeans that he did not wear that blouse.B3. If a mouse did not get agift from Gauss, it must be that he hasn’t tried on all
the blouses yet.
Statement C. If Statement A is true then Statement B is true.
C1. In every house there is a mouse that never wore a blouse,but got a gift from Gauss.
C2. Every house has at least 3 mice, but under no conditionwould Gauss give something to a mouse.
C3. There is a mouse in my house that likes to wear a silverblouse and got some cookies from my spouse. (My name’s JohannaGauss).
