1. (6 marks)
Let
A
=
{
1
,
2
,
3
,
4
,
5
}
,
B
=
{
3
,
6
}
, and let
E
be the set of even integers. Find
(a)
A
∪
B
=
{
1
,
2
,
3
,
4
,
5
,
6
}
(b)
A

E
=
{
1
,
3
,
5
}
(c)
A
∩
E
=
{
2
,
4
}
(d)
P
(
B
)=
{∅
,
{
3
}
,
{
6
}
,
{
3
,
6
}}
(e)
B
× {∅}
=
{
(3
,
∅
)
,
(6
,
∅
)
}
(f) (
B
⊆
A
)
∨
(
B
⊆
E
) = False (or F)
2. (3 marks)
Let
p
and
q
be statements. Use a truth table to show that (
p
→
q
)
↔
(
∼
q
→∼
p
)i
sa
tautology.
pq
p
→
q
∼
q
∼
p
∼
q
→∼
p
(
p
→
q
)
↔
(
∼
q
p
)
TT
TF
F
T
T
FT
F
F
T
T
T
T
FF
T
T
T
3. (6 marks)
Let
F
(
x
) mean that student
x
is a frosh, and let
C
(
x, y
) mean that student
x
is enrolled in
class
y
, where the domain of
x
is the set of all students at U of T and the domain of
y
is the
set of all classes at U of T.
Express the following statements in simple English.
(a)
C
(Alice
,
ECE190)
Alice is in ECE190.
(b)
∀
x
(
F
(
x
)
→
C
(
x,
FRS101))
All frosh are in FRS101.
(c)
∃
x
∃
y
∀
z
((
x
±
=
y
)
∧
(
C
(
x, z
)
↔
C
(
y,z
)))
There are two students enrolled in the same set of classes.
Write the following statements using predicates and quanti±ers, the negation symbol
∼
, and
the logical connectives
∧
,
∨
,
→
,
↔
and
⊕
. If negations are used, express the statements so
that no negation symbol is to the left of a quanti±er.
(d) Bob is a frosh, but he is not enrolled in CIV101.
F
(Bob)
∧∼
C
(Bob,CIV101)
(e) Being a frosh is a necessary condition for any U of T student to be enrolled in ECE190.
∀
x
(
C
(
x,
ECE190)
→
F
(
x
))
(f) It is not the case that every student is enrolled in at least one course.
∃
x
∀
y
∼
C
(
x, y
)
2