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].