Problem
1
2
3
4
5
6
7
8
9
Total
Mark
MACM 101
Midterm Test
November 17, 2010
Last Name
First Name and Initials
Student No.
Tutorial section or TA name
NO AIDS allowed. Answer ALL questions on the test paper. Use backs of
sheets for scratch work.
Total Marks: 100
1.
Give a deﬁnition of an equivalence relation.
[10]
An equivalence relation is a reﬂexive, symmetric transitive relation.
2.
Give a deﬁnition of a onetoone function.
[10]
Function
f
is onetoone if
∀
a
∀
b
(
f
(
a
) =
f
(
b
)
→
a
=
b
).
3.
Let
S
(
x
)
means “
x
is a student”,
F
(
x
)
means “
x
is a faculty
member”,
A
(
x,y
)
means “
x
has asked
y
a question”, where the
domain consists with all people associated with SFU. Formulate
in logic: There is a faculty member who has never been asked a
question by a student.
∃
x
(
F
(
x
)
∧ ∀
y
(
S
(
y
)
→ ¬
A
(
y,x
)))
[10]
4.
Use mathematical induction to prove
n <
2
n
for all positive in
tegers
n
.
[15]
(a) Basis case: Check the property for
n
= 1: 1
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 Summer '08
 PEARCE
 Mathematical Induction, Natural number, Equivalence relation, Transitive relation, Partially ordered set, positive integers

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