# Chapter 2 Questions.

## Predicate logic.

### Question 1

In this question, assume the following predicate and constant symbols:

W(x,y) : x wrote y
L(x,y) : x is longer than y
N(x) : x is a novel
h : Hardy
a : Austen
j : Jude the Obscure
p : Pride and Predjudice

Given these specifications, which of the predicate logic formulas below represent the sentence, 'Hardy wrote a novel which is longer than any of Austen's' in predicate logic?

### Question 2

In this question, assume the following predicate and constant symbols.

W(x,y) : x wrote y
L(x,y) : x is longer than y
h : Hardy
a : Austen
j : Jude the Obscure
p : Pride and Predjudice

Which of the following represents the sentence, ''Jude the Obscure' is not longer than 'Pride and Predjudice'', in predicate logic?

### Question 3

Working with a unary predicate sysmbol, P, a binary predicate symbol Q and a unary function sysmbol f, which of the following formulas are satisfied in the model M, given by,

A = {a,b,c,d};
PM = {a,b}
QM = {(a,b), (b,b), (c,b)}
fM (a) = b, fM (b) = b, fM (c) = a, and fM (d) = c

### Question 4

Which of the following sets of sentences is satisfiable? (Recall that a "sentence" is a predicate logic formula that has no free variables.)

### Question 5

Which of the following semantic entailments are valid in predicate logic?

### Question 6

In this question assume the specifications

F(x,y) : x is the father of y
M(x,y) : x is the mother of y.

Which of the formulas in predicate logic below express the sentence 'Everybody has a mother'?

### Question 7

In this question assume only the specifications

F(x,y) : x is the father of y
M(x,y) : x is the mother of y.

Which of the formulas in predicate logic below express the sentence 'Anyone who is a father is not a maternal grandmother'?

### Question 8

For this question and the following one, assume the specifications below:
The predicate symbols

S(x,y) : x is y's sister
B(x,y) : x is y's brother
H(x,y) : x is y's husband and
O(x,y) : x is older than y

and the constant symbols

j : John,
c : Carl, and
m : Monique.

Which of the following formulas express the sentence 'Carl is Moniques brother-in-law'?

### Question 9

Assuming the same specifications as for Question 8, which of the following sentences are expressed by the predicate logic formula
x y (S(x,j) S(y,j) O(x,y))?.

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