(30 pts) Three chests lie before you. Each chest will either only tell the truth or lie. Only one chest will contain gold.
Chest 1: The gold is in this chest.
Chest 2: The gold is not in this chest.
Chest 3: The gold is not in chest 1.
Use the following to help formulate proof of who is telling the truth and where the gold is. A closed system is one in which the result is concluded. You will convert the logical statements into a Boolean algebraic expression, whose result should be 1 to prove your findings.
x = starting point conditions met
p = hypothesis
q = conclusion
Lets tie in the propositions so that they form a closed system: $x \land (p \implies q)$
a. Define the variables, atomic propositions, and quantifiers as logical representations.
b. Use the quantifiers to simplify and build atomic propositions (you might need some xors) in terms of logical representation.
c. What is the complex statement we intend to prove?
d. Create new variables for your three propositions obtained to describe the 3 chests, then write a boolean expression in the form z = 1 in terms of x and y. You should be able to arrive to this based on proper construction of proposition statements for the 3 chests.
e. Simplify the boolean expression above and justify steps, this will help you determine the value for y, which you can use to rationalize x.
f. Based on the answer in e, determine where the gold is.
g. Present the overall proof.
