(b) Gandalf fools everyone.
(c) The only person Frodo can fool is himself.
_________________________________________________________________
5. (5 pts.)
Determine the truth value of each of the
following statements if the universe of discourse of each
variable is the set of natural numbers,
= {0,1,2,...}.
(a) (
∀
x)(
∃
y)(x + y = y)
(b) (
∃
x)(
∀
y)(x + y = y)
_________________________________________________________________
6. (5 pts.)
If A is a countable set and B is an uncountable
set, must A  B be countable? Briefly explain.
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TEST1/MAD2104
Page 3 of 4
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7. (15 pts.) Suppose A = {
∅
,{
∅
},2} and B = {2,3}. Then
A
∪
B =
A × B =
P(A) =
_________________________________________________________________
8. (5 pts.) Suppose A and B are subsets of a universal set U.
Show that A  B =
∅
→
A
⊆
B.
[Hints:
(1) x
ε
A
→
x
∉ ∅ →
... .
(2) A  B =
∅ → ∀
x( x
ε
A  B
→
x
ε ∅
). The
contrapositive of the implication within the parentheses here is
useful in dealing with the ellipsis in hint #1.]
_________________________________________________________________
9. (5 pts.) If f:X
→
Y is a function, f
1
may be used to denote
two quite different things. What are they? [Use complete
sentences.]
TEST1/MAD2104
Page 4 of 4
_________________________________________________________________
10. (15 pts.) Suppose that f:
→ Ζ
is the function defined by
the formula f(x)= x , and suppose that A = {x
ε
 3
≤
x
≤
3} and
B = {x
ε
 1 < x
≤
π
}. Using appropriate notation, give each of
the following.
A  B =
f(B) =
f
1
({1,3}) =
_________________________________________________________________
11. (5 pts.)
What is the value of the following sum of terms of
a geometric progression? [Hint: You may wish to reindex the
varmint.]
8
∑
2
j
=
j=1
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12. (5 pts.)
Suppose g:A
→
B and f:B
→
C are functions. Prove
exactly one of the following propositions. Indicate clearly which
you are demonstrating.
(a) If f g:A
→
C is injective, then g is injective.
(b) If f g:A
→
C is surjective, then f is surjective.
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 Spring '08
 STAFF
 Logic, pts, Countable set

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