January:23(Fri)
Due January 30 (Friday) at Lecture
-
Read
our textbook's
Chapter 1.
Please do
all of these exercises in teams of (at most) two.
Such teamwork earns a 5% bonus for you.
Use
ProofBuilder
to (re)do the following exercises
(earlier assignments asked you to do many of these
by hand).
- 1.1.12a,b,d
- 1.1.14d
- 1.2.12a
(Again) operate like Example 8 on page 27 of our textbook,
by selecting appropriate parts of formulas and
invoking "Apply equation or equivalence"
from the "Deduction" menu.
- 1.2.22
(Again) operate like Examples 6-7 on page 26 of our textbook,
as follows:
-
Enter the following formula as the theorem:
(p → q) ∧ (p → r) ≡ p → (q &and r)
By the way,
note
that
you can copy this text from this Web page and paste
what you've copied
into ProofBuilder.
-
Invoke "Transform one side to other"
from the "Deduction" menu.
-
Then,
select appropriate parts of formulas and
invoke "Apply equation or equivalence"
from the "Deduction" menu.
- 1.2.24
Do this exercise like 1.2.22 (above).
- 1.3.13a,b,c
After typing each formula
as the theorem
in ProofBuilder,
invoke "Assign value(s) to variable(s)"
from the Deduction menu.
(Actually, for item a, you'll have nothing to do
after you enter the formula.)
- 1.3.35b,c
Note the solutions in the back of the book.
Since the intention is to prove that each formula is false,
type "¬" at the beginning when you enter
each of these formulas
for the theorem in ProofBuilder; e.g.:
¬ ∀x(x > 0 ∨ x < 0)
By the way,
note
that
you can copy this text from this Web page and paste
what you've copied
into ProofBuilder.
After typing each formula
(with "¬")
as the theorem
in ProofBuilder,
select the quantifier "∀x"
and then
invoke "Assign value(s) to variable(s)"
from the Deduction menu.
(Note the solutions in the back of the book.)
- 1.3.12
If you believe that an item's formula is false,
then
type "¬" at the beginning when you enter the formula
for the theorem in ProofBuilder.
E.g. with Exercise 1.3.11,
item c's formula "P(2)" is false,
so one would enter "¬P(2)"
as the theorem in ProofBuilder;
and similarly
Exercise 1.3.11's
item f's formula "∀x P(x)" is false,
so one would enter "¬ ∀x P(x)"
as the theorem in ProofBuilder.
Note also that there's a thin space character, " ",
between "∃x"
or "∀x"
and "Q(x)".
To enter this thin space character,
press the alt key
while you
type "_"
or "|".
Then,
for each of the items of this exercise,
after entering the appropriate formula as the theorem
in ProofBuilder,
invoke "Add presuppositions"
from the Management menu
and
enter the following formula as a presupposition —
you can copy this text from this Web page and paste
what you've copied
into ProofBuilder:
∀x[Q(x) means x + 1 > 2x]
Then
apply this presupposition to the "Q()"
expression in the theorem
by
selecting both of them and
invoking "Apply equation or equivalence"
from the Deduction menu.
Then further,
for items d-g,
select the quantifier
and assign a value to the variable
by
invoking "Assign value(s) to variable(s)"
from the Deduction menu.
- 1.3.36a,c
Type "¬" at the beginning when you enter each formula
for the theorem in ProofBuilder.
- 1.4.27a,c,d,e
Do
each proof by removing quantifiers and/or assigning values to variables.
- 1.4.28a,c
- 1.4.30