Question

5. Consider the predicates m(X,Y) = X is a member of Y, s(X) = X is a skier, mc(X) = X is a mountain climber, and l(X, Y) = X likes Y. You are given a set of axioms written in first-order logic (in the right column below) whose meanings are given in plain English sentences (left column below): Tony, Bill and John are members of the Alpine Club. m(t,alp) ^ m(b,alp) ^ m(j,alp) All Alpine Club members are skiers or mountain climbers. (\forall X)(m(X,alp) \to s(X) \lor mc(X)) No mountain climber likes rain. (\forall X)(mc(X) \to \neg l(X,rain)) All skiers like snow. (\forall X)(s(X) \to l(X,snow)) Bill dislikes whatever Tony likes. (\forall X)(l(t,X) \to \neg l(b,X)) Bill likes whatever Tony dislikes. (\forall X)(\neg l(t,X) \to l(b,X)) Tony likes rain and snow. l(t,rain) \land l(t,snow) a. Express the conjecture \"There is someone who is a member of the Alpine Club and who is a mountain climber but not a skier.\" in first order predicate calculus using the same predicates. b. Converting the above information into appropriate clauses, show how resolution would produce a refutation tree that proves that the conjecture in (a) follows from the given axioms. Be sure to show all substitutions adjacent to the arcs of the derivation tree.

          5. Consider the predicates m(X,Y) = X is a member of Y, s(X) = X is a skier, mc(X) = X is a
mountain climber, and l(X, Y) = X likes Y. You are given a set of axioms written in first-order
logic (in the right column below) whose meanings are given in plain English sentences (left
column below):
Tony, Bill and John are members of the Alpine Club.
m(t,alp) ^ m(b,alp) ^ m(j,alp)
All Alpine Club members are skiers or mountain climbers.
(\forall X)(m(X,alp) \to s(X) \lor mc(X))
No mountain climber likes rain.
(\forall X)(mc(X) \to \neg l(X,rain))
All skiers like snow.
(\forall X)(s(X) \to l(X,snow))
Bill dislikes whatever Tony likes.
(\forall X)(l(t,X) \to \neg l(b,X))
Bill likes whatever Tony dislikes.
(\forall X)(\neg l(t,X) \to l(b,X))
Tony likes rain and snow.
l(t,rain) \land l(t,snow)
a. Express the conjecture \"There is someone who is a member of the Alpine Club and who is a
mountain climber but not a skier.\" in first order predicate calculus using the same predicates.
b. Converting the above information into appropriate clauses, show how resolution would
produce a refutation tree that proves that the conjecture in (a) follows from the given axioms. Be
sure to show all substitutions adjacent to the arcs of the derivation tree.

5. Consider the predicates m(X,Y) = X is a member of Y, s(X) = X is a skier, mc(X) = X is a
mountain climber, and l(X, Y) = X likes Y. You are given a set of axioms written in first-order
logic (in the right column below) whose meanings are given in plain English sentences (left
column below):
Tony, Bill and John are members of the Alpine Club.
m(t,alp) ^ m(b,alp) ^ m(j,alp)
All Alpine Club members are skiers or mountain climbers.
(∀X)(m(X,alp) →s(X) mc(X))
No mountain climber likes rain.
(∀X)(mc(X) →l(X,rain))
All skiers like snow.
(∀X)(s(X) →l(X,snow))
Bill dislikes whatever Tony likes.
(∀X)(l(t,X) →l(b,X))
Bill likes whatever Tony dislikes.
(∀X)(l(t,X) →l(b,X))
Tony likes rain and snow.
l(t,rain) l(t,snow)
a. Express the conjecture T̈here is someone who is a member of the Alpine Club and who is a
mountain climber but not a skier.ïn first order predicate calculus using the same predicates.
b. Converting the above information into appropriate clauses, show how resolution would
produce a refutation tree that proves that the conjecture in (a) follows from the given axioms. Be
sure to show all substitutions adjacent to the arcs of the derivation tree.

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