Reference no: EM132431080

Any help would be greatly appreciated.

1) If the predicate loves(X,Y) is interpreted as "person X loves person Y," then what is the best interpretation of the logical expression (∀X)(∃Y)[loves(X,Y)]? Note: the correct answer may involve logically rewriting the expression into an equivalent form.  

a) Each person is loved by at least one person.

b) Everybody loves everybody.

c) There is no one whom no one loves.

d) Each person has at least one person that they love.

2) Here is an expression of predicate logic, where all the variables are X. The subscripts are not part of the expression, and are there just so we can discuss particular occurrences of the variable X.

(∀X1)[(∃X2)(p(X3) → X4=10) OR X5=20 OR (∃X6)(q(X7) → X8=30)]

According to the binding rules for predicate-logic expressions, certain of these instances of X refer to the same variable, and others refer to different variables. Find all groups of occurrences of X that are the same variable and, in the list below, identify one pair of occurrences that refer to the same variable.

a) X1 and X3

b) X2 and X5

c) X7 and X8

d) X5 and X6

3) If we imagine that the predicate loves(X,Y) means "X loves Y," then the following sentence:

(∀X)(∀Y){(∃Z)[loves(Y,Z)] → loves(X,Y)}

can be interpreted as saying "everybody loves a lover." We can form the negation of this sentence, i.e., "not everybody loves a lover," simply by putting a NOT in front of the sentence. However, we can also make a number of transformations on the negation. For example, there are rules for pushing a NOT through an existential or universal quantifier, and through an "implies" operator →. In some situations, it is also possible to commute the order of quantifiers, move quantifiers inside or outside operators like →, and commute the order of certain operands, e.g., of and AND operator. Consider some of the changes that could be made to the negated expression:

NOT{(∀X)(∀Y)[(∃Z)(loves(Y,Z)) → loves(X,Y)}

to derive other equivalent expressions. Identify the one in the list below that is NOT equivalent to the expression above.

a) (∃X)NOT{(∀Y)[(∃Z)(loves(Y,Z)) → loves(X,Y)}

b) (∃X)(∃Y){NOT loves(X,Y) AND (∃Z)[loves(Y,Z)]}

c) NOT{(∀Y)(∀X)[(∃Z)(loves(Y,Z)) → loves(X,Y)}

d) NOT{(∀X)(∀Y)[(∃Z)(loves(Y,Z)) AND NOT loves(X,Y)}