First Order Logic Assignment | Mathematic Assignment Help

Question 1

Consider the following truth function:

To enter logical symbols on the keyboard, use:

not ¬	-, ~, not
and ?	/\, &, and
or ?	\/, |, or
if then ?	->, >, only if
if and only if ??	<->, <>, if and only if
contradiction ?	!?, _|_
universal quantifier ?	A, @
existential quantifier ?	E, 3

A	B	C	@(A,?B,?C)
T	T	T	F
T	T	F	T
T	F	T	F
T	F	F	F
F	T	T	F
F	T	F	T
F	F	T	F
F	F	F	T

Give a sentence in standard form (formula) expressing @.

1. @(A,B,C) in Standard Form:
2. @(A,B,C) using each of ~, /\, > only once:

 

Question 2

	Please formalize each of the following sentences into FOL following the key and domain. Note that in this town, there are only two professions: logician and geologist.
Domain:	People in the same town as Doreen
W(x,?y):	x works with y
P(x,?y):	x is a parent of y
S(x,?y):	x is the sibling of y
O(x,?y):	x is younger than y
L(y):	y is a logician
G(y):	y is a geologist
d:	Doreen
j:	Jenifer
h:	Heather

1. Everyone is older than their children.
2. Heather's only parent is a geologist.
3. Jenifer is the only child of Doreen and Heather.
4. Doreen works with someone the same age as Jenifer.
5. No one is in the same profession as their parents.
6. If someone is an only child, then they are not a logician, but they have multiple children.
7. Heather and all of her grandparents are geologists and she has at least two grandparents

 

Question 3

	Please formalize the following sentences into FOL using the key and domain provided.
Domain:	Everything
U(x,?y,?z):	x understands y at least as well as z
B(x,?y):	x bores y
L(x,?y):	x listens to y
S(x,?y):	x is a student of y
W(x,?y):	x wastes y's time
P(x):	x is a professor
R(x):	x is unpopular

 

1. If a professor doesn't listen to any of their students, then they are wasting their students' and their own time.
2. Professors only bore those of their students who bore some of their professors.
3. Some professors understand somethings worse than some of their students.
4. Unpopular students either waste some other student's time or listen to all of their own professors.
5. Just because a professor bores some of their students doesn't imply that they are unpopular; but when none of a professor's students listens to them, that does imply that they are unpopular.
6. If something is understood equally well by all of a student's professors, then that student doesn't undersand it as well as their professors.

 

Question 4

	Determine the truth values of the following sentences on the given interpretation using their usual extensions:
Domain:	Actual People
B(x)	is true of people under 18 years old
C(x)	is true of people over 40 years old
D(x,?y)	is true of all pairs ?<!--?y>, that own the same boat
F(x,?y)	is true of all pairs ?<!--?y>, each of whom doesn't own a boat

 

1. ?x?y(B(y)???F(x,?y))

Write whether the formula above is true or false in this interpretation:

2. ?x?y?z[(D(x,?y)???F(y,?z))???D(x,?z)]

Write whether the formula above is true or false in this interpretation:

3. ?w(C(w)????xD(x,?x))

Write whether the formula above is true or false in this interpretation:

4. ?x?y(F(x,?y)???C(x))

Write whether the formula above is true or false in this interpretation:

5. ?x?y(D(x,?y)????F(x,?y))

Write whether the formula above is true or false in this interpretation:

6. ?xC(x)???(?x?yF(x,?y)????yB(y))

Write whether the formula above is true or false in this interpretation:

7. ¬?y(C(y)???B(y))

Write whether the formula above is true or false in this interpretation:

8. ?xB(x)????xC(x)

Write whether the formula above is true or false in this interpretation:

9. ?x?y(D(x,?y)???¬F(y,?x))

Write whether the formula above is true or false in this interpretation:

10. ?x?y(C(x)???F(y,?x))

Write whether the formula above is true or false in this interpretation:

Question 5

Consider the following argument:

?x(B(x)???(C(x)????yD(x,?y)))

?x(B(x) ???¬?zD(x,?z))

???x(B(x)????yD(x,?y))

Provide a counterexample with two objects

1. ?x?y¬x=y

Domain:

B(_):

C(_):

D(_,_):

 

Provide a counterexample where B(x) is true of at least two objects.

2. ?x?y(¬x=y ? (B(x) ? B(y)))

Domain:

B(_):

C(_):

D(_,_)

• Due date: July 30, 8 am, GST