Given the following multiplyquantified statement z Pz Qz y Q

Given the following multiply-quantified statement z (P(z) Q(z)) y Q(y) R(z)

a. Parenthesize the expression so it is explicit which operands go with which operands and so the scope of each quantifier is a parenthesized expression.

b. For each occurrence of each variable, indicate if it is bound or free?

c. Rename the variables so that distinct names are used for each distinct variable

Solution

Given z (P(z) Q(z)) y Q(y) R(z)

a. it can be written as (P(z)  z ^ Q(z)  z) v  y   

which is said as  Q(y) R(z)

where P(z) belongs to for all z and Q(z) belongs to for all z or all values of y which can be given as Q(y)--> R(z)

b. P(z) is bounded over z

Q(z) is bounded over z

P(z) ^ Q(z) is bounded over z

y is not bounded

c. (P(z)  z ^ Q(z)  z) v  y

renaming the variables writing z as a and y as b

(P(a)  z ^ Q(a)  z) v  b