Introductory Chapter : Mathematical Logic, Proof and Sets 11
Exercise I(a)
1.
Determine the elements of the following sets:
(a) A

 x 2 x
 
0

(b) B

 x
 x

1
 x

2
 
0

(c)
(e)
C
E



 x x x x
2
2

2 x
 
9 0


(d)
(f)
D

 x
F

 x x x
2 
4 x
 
0

is a prime number less than 10

2.
Determine the elements of the following sets:
(a)
(c)
A
C



 x x

 
 x x


1

 x x
3


0

0

(b)
(d)
B
D



 x x


(e) E

 x
 x

0

(f) F

 x
 x
2 x
 
0

 x
  x

0


0

3.
Write the following statements in set notation:
(a) The set of negative real numbers.
(b) The set of positive integers.
(c) The set of real numbers between 0 and 2 excluding 0 and 2.
(d) The set of rational numbers less than 1.
(e) The set of natural numbers which are multiples of 10.
Consider the following Venn diagram for questions 4 to 7:
A B
Fig 12
4.
Shade in the following regions.
(a) c
A (b) c
B (c)

A

B
 c
(d)
U

A

B
 c
5.
Shade the following regions of the Venn diagram of Fig 12:
(a) \ (b)
  c
B A (c)

A B
 c
(d)
    c
6.
Show that for the sets in Fig 12 we have the result

A

B
 c 
A c 
B c
.
7.
By shading the Venn diagrams of Fig 12 show that

A

B
 
A

B
  
B
8.
Let
B

U
 
1, 2, 3, 4, 5, 6, 7, 8, 9


1, 3, 5, 7

be the universal set and
. Determine the members of the following sets:
A
 
2, 4, 6, 8

,
(a) A

B
(f) A

B
(b) A

B (c) c
A (d) c
B (e) \

Introductory Chapter : Mathematical Logic, Proof and Sets
9.
Let
C

 n

A

 n

(a) A B C (b) A
 
C
(e) \ (f) n
2 , 1

B

 n
 n 8

and
5

. Determine the elements of the following sets:

A

B

\ C
(c)
(g)
A
 
B

C


\

\ C
(d) \
12
For the remaining questions use the Venn diagram of Fig 13.
A
C
B
Fig 13
U
10. Shade in the following sets:
(a) A
 
\

(b)

A

B
 
A

C

What do you notice about your results?
11. Shade in the following sets:
(a)

\

\ C (b)

\

\ A
What do you notice about your results?
Brief Solutions to Exercise I(a)
1. (a)
(e)
A



1
2


E
 
3, 3

(b)
(f)
B

F
 
2, 3, 5, 7

(c) C
  
(d) D

1, 5

2. (a) A
  
(b) B
 
(c) C
 
(d)
or (c)
 x
D

1
3
,

5

(e) E
 
3. (a)
(d)
 x
 x

 x
(f) x

0

(b)

1

(e)
F
 x
  
 x

0


10 n n



0
 
2
8. (a)
(c)
A B
A c 

1, 2, 3, 4, 5, 6, 7, 8


1, 3, 5, 7, 9

(d)
(e) \
  
2, 4, 6, 8

(f)
B c  
2, 4, 6, 8, 9

A B
(b) A B

1, 2, 3, 4, 5, 6, 7, 8

9. (a)
(b)
A B C
A B C

2, 4, 6, 8, 10, 12, 14, 16, 20, 32


4, 8, 16

(c) A
 
B

C
 
2, 4, 8, 16

(d)
(f)
\
 
2, 32

(e)

A

B

\ C
 
2, 6, 10, 14, 32

\
 
2, 6, 10, 14

(g)

\

\ C
  
10. The same regions are shaded therefore
11. Different sets, that is
A
 
\
 
A

B
 
A

C

.

\

\ C
 
\

\ A [Not Equal].
