ICS.6D, Fall 2007, Midterm, November 5, 2007,
SOLUTIONS
1)
True, False, or Depends?
[ 48 pts
= 16 items x
3pts each ]
For each statement mark whether it’s True, False, or it depends on the instantiation of the
variables and/or the definition of the predicates:
a.
If (p and
¬
q) then p
True, because (p and r) implies p for any r
.
b.
If p then (10=5 and a=a)
Depends, because it’s true whenever p ≡ F and
false whenever p ≡ T, because (10=5 and a=a) ≡ F, and (p→F) ≡ (
¬
p).
c.
(p
↔
¬
p)
False, because it’s false for all p.
d.
(p
↔
¬
p) is a contradiction
The answer to (d) is “True” because the answer to
(c) is “False”.
In other words, (p
↔
¬
p) is a contradiction because it’s a
statement that’s false for all values of the variable p.
e.
(
2200
x
¬
A(x))
if
(
¬ 5
x A(x))
True, because these two statements are equivalent
for all predicates A.
Note that implicitly both quantifiers are over the same
universe (which was not stated but was to be assumed).
f.
{1,4}
⊂
({1,2,3,4,5} – {2,3})
True, because {1,4} is a proper subset of {1,4,5}
g.

{1,1,2,2,3} x {1,1,1}

= 3
True, because {1,1,2,2,3}=3, {1,1,1}=1, and 3*1=3
h.

P({1,1,2})

= 2
False, because {1,1,2}=2, so P({1,1,2}=2
2
=4
i.
Ø
∈
{ Ø }
True, just a case of a
∈
{a} for some a.
j.
Ø
⊂
{ Ø }
True, because Ø
⊆
A
for all A and Ø ≠ {Ø}
k.
S
⊂
S
False, because S=S for all S.
l.
(A
∪
B)
∩
C = (A
∩
C)
∪
(B
∩
C)
True.
It’s one of the distribution laws and it
can be quickly checked using basic definitions of union or intersection or using
a Venn Diagram.
m. 2
2
•3
2
= gcd( 2
2
•3
3
, 2
2
•3
1
)
False, because the gcd here is 2
2
•3
1
n.
gcd(a,b) = gcd(b mod a, a)
True for all integers (a,b) s.t. a≠0.
The Euclidean
Algorithm for fast gcd computation is based on this fact for b>=a.
However it’s
also true for a>b because if a>b then (b mod a) = b, and it’s true for b=a because
gcd(0,a)=a if a≠0.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Jarecki
 Integers, Quantification, Euclidean algorithm, implicit domain

Click to edit the document details