Mathematics for Biotechnology and Data Science
Chapter 3: Application of Set Theory
Chapter 6: MAPPING OR FUNCTIONS
Chapter 8: Inverse of a Function
4. VENN DIAGRAMS
Sometimes pictures are very helpful in our thinking. First of all, a Swiss mathematician Fuler gave an idea to represent a set by the points in a closed curve. Later on, British mathematician Venn brought this idea to practice. That is why the diagrams drawn to represent sets are called Venn-Euler diagrams or simply Venn diagrams. In Venn diagrams the universal set U is represented by points within a rectangle and its subsets are represented by points in closed curves (usually circles) within the rectangle. If a set A is a subset of a set B, then the circle representing A is drawn inside the circle representing B. If A and B are not equal but they have some common elements, then to represent A and B we draw two intersecting circles. Two disjoint sets are represented by two non-intersecting circles.
4.1 Operations on Sets through Venn diagram
In this section, we shall introduce some operations on sets to construct new sets from given ones.
Union of Sets: Let A and B be two sets. The union of A and B is the set of all those elements which belong either to A or to B or to both A and B.
We shall use notation 
(reads as “A union B”) to denote the union of
A and B.
Thus,
.
Clearly, 
.
And,
.
In above figure, the
shaded part represents .
It is evident form the Venn
diagram that 
.
Example 1 If
A = {1, 2, 3} and B = {1, 3, 5, 7}, then 
. Draw Venn diagram.
Example 2 If
and 
then
Intersection of Sets: Let A and B be two sets. The intersection of A and B is the set of all those elements that belong to both A and B.
We shall use notation 
(reads as “A intersection B”) to denote the
intersection of A and B.
Thus,
.
Clearly, 
.
In above figure, the
shaded region represents .
Evidently
.
Example 3 If
A = {1, 2, 3, 4, 5} and B = {1, 3, 9, 12}, then
. Draw Venn diagram.
Example 4 If
and 
then
Draw
Venn diagram.
Disjoint Sets:
Two
sets A and B are said to be disjoint, if 
. If then A and B are said to be intersecting or
overlapping sets.
Example 5 Let A = {1, 2, 3, 4, 5, 6}, B = {7, 8, 9, 10, 11} and C = {6, 8, 10, 12, 14}, then A and B are disjoint sets, while A and C are intersecting sets.
Difference of Sets:
Let
A and B be two sets. The difference of A and B, written as 
, is the set of all
those elements of A which do not belong to B.
Thus,
or 
.
Clearly, 
.
In the following
figure, the shaded part represents 
.
Similarly, the
difference 
is the set of all those elements of B that do
not belong to A
i.e 
.
In the following figure
the shaded part represents
Example 6 If
A = {2, 3, 4, 5, 6, 7} and B = {3, 5, 7, 9, 11, 13}, then 
and 
.
Symmetric Difference of
Two Sets: Let A and B be two
sets. The symmetric difference of sets A and B is the set 
and is denoted by 
.
Thus, 
The
shaded part in the following figure represents the
Example 7 If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 6, 7, 8, 9} then
and so = {2, 4, 9}
Example 8 If
and 
then
, 
and
.
Complement Sets:
Let
U be the universal set and let A be a set such that 
. Then the complement
of A with respect to U is denoted by 
or 
or 
and defined the set of all those elements of U
which are not in A.
Thus 
.
Clearly 
The shaded part in the
following figure represents the
Example 10 Let
the set of natural numbers N = {1, 2, 3, 4…} be the universal set and let A =
{2, 4, 6, 8…} then 
. Draw Venn diagram.
Example 11 If
U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 3, 5, 7, 9} then 
. Draw Venn diagram.
EXERCISE
|
Q:1 If A and B are two sets such that number of elements in A is 24, number of elements in B is 22 and number of elements in both A and B is 8, Find: i.
ii.
iii.
Also draw Venn diagram Ans (i) 38 (ii) 16 (iii) 14 |
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Q:2 If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14} then represent these four sets in Venn diagram.
|
|
Q:3
Let
Find
(1) Ans
(1)
B (2) C (3) D (4) |
|
Q:4 If A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12 , 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20} then Find
(1) A Ans (1) {3, 6, 15, 18, 21} (2) {3, 15, 18, 21} (3){3, 6, 12, 18, 21} (4){4, 8, 16, 20} (5){2, 4, 8, 10, 14, 16} (6) {5, 10, 20} (7) {20} (8){4, 8, 12, 16} |
|
Q:5 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6} then Find
(1) Ans (1) A (2) {1, 3, 5, 7, 9} (3){2, 4, 5, 6, 7, 8, 9} (4) {5, 7, 9} (5) A |
|
Q:6 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7} then verify
that (1) |
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