Question

QUESTION 1. For each of the following, state whether the occurrence of the variable x occurs bound, or free (i.e. unbound), both, or neither.

1. ∃xCube(x)

2. ∀xCube(a) ∧ Cube(x)

3. ∀x((Cube(a) ∧ Tet(b)) → ¬Dodec(x))

4. ∃yBetween(a,x,y)

5. ¬∀x¬(¬Small(d) ∧ ¬LeftOf(c,x))

QUESTION 2. Correctly label each of the following strings of symbols as a sentence, or well-formed formula (but not a sentence), or neither.

1. Fx ∧ Gy

2. ∃bFb

3. ∃z(Fz → Gb)

4. ∀xFc

5. ∀yFy ∨ ¬Fy

6. ¬∃¬xGx

Help me please with these questions thank you

Answer 1

QUESTION 1:

A free variable is one which has no quantifiers (there exists, for all, etc.) associated with it. A variable that is not free is said to be bound.

Answers:

(1) Bound [Due to presence of the quantifier ∃.]

(2) Both [Due to presence of the quantifier ∀ in first part and lack of any quantifier in second part (the part after ∧).]

(3) Bound [Due to presence of the quantifier ∀.]

(4) Free [Due to lack of any quantifier associated with the variable x.]

(5) Bound [Due to presence of the quantifier ∀.]

QUESTION 2:

A well-formed formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Sentences are closed formulas, that is formulas with no free variables.

Answers:

(1) Well-formed formula [Since the strings of symbols has correct syntax and the variables are free.]

(2) Sentence [Since the strings of symbols has correct syntax and the variable b is not free.]

(3) Well-formed formula [Since the strings of symbols has correct syntax and the variable b is free.]

(4) Well-formed formula [Since the strings of symbols has correct syntax and the variable c is free.]

(5) Well-formed formula [Since the strings of symbols has correct syntax and the variable y is free in the second part (the part after ∨).]

(6) Neither [Since the strings of symbols does not have correct syntax (¬∃¬x is incorrect, ¬∃x would have been correct).]