Question

Problem 1. The purpose of this problem is to practice the use of logic operations and quantifiers. For each Statement X below determine if each of the three statementsX1, X2, X3 that follow it satisfy 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 must be false.

Statement A. In every house there is a mouse.

A1. There is no house without a mouse.A2. There exists a house without a mouse.A3. Mice don’t live in houses.

Statement B. For every mouse there is a blouse, such that if the mouse wears the blouse he’ll get a gift from Carl Friedrich Gauss.

B1. There is a mouse that can wear any blouse, but still won’t get a gift from Gauss.

B2. There are no mice for which there does not exist a special blouse, such that if the mouse is not getting a gift from Gauss it means that he did not wear that blouse.B3. If a mouse did not get a gift 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 condition would Gauss give something to a mouse.

C3. There is a mouse in my house that likes to wear a silver blouse and got some cookies from my spouse. (My name’s Johanna Gauss).

Answer 1