With respect to the interpretation which:
which of the following are valid:
Problem 1
•• x < y à (x - z) < (y -z)
valid
Problem 2
•• x < y v y < x
not valid
Problem 3
•• (x < y v x == y) ^ (y < x v x == y) à (x == y)
Assuming that equal and distinct are predicates such that equal(X, Y) is true when X and Y are the same, and distinct(X, Y) is true when they are different, express rules with respect to the preceding Prolog database that define:
Problem 4
•• sibling(X, Y) means X and Y are siblings (different people having the same parents)
sibling(X, Y) :- parent(Z, X), parent(Z, Y).
Problem 5
•• cousin(X, Y) means that X and Y are children of siblings
cousin(X, Y) :- parent(U, X), parent(V, Y), sibling(U,V).
Problem 6
••• uncle(X, Y) means that X is the sibling of a Z such that Z is the parent of Y, or that X is the spouse of such a Z.
uncle(X, Y) :- sibling(X, Z), parent(Z, Y).
uncle(X, Y) :- husband(X, Z), parent (Z, Y).
Problem 7
••• brother_in_law(X, Y) [consult a dictionary for the definition]
Which of the following are valid for the natural numbers interpretation?
Problem 8
• ($x) (x != x)
not valid
Problem 9
• (œy)($x) (x != y)
valid
Problem 10
• (œy)($x) (x = y)
valid
Problem 11
•• (œy)(œx)(x = y) v (x > y) v (x < y)
valid
Problem 12
•• (œy) [ (y = 0) v ($x) (x < y) ]
valid
Problem 13
•• (œy) [ (y = 0) v ($x) (x > y) ]
valid