Let us understand the use of Venn diagrams from the
following example.
e.g. A = { 1, 2, 3, 4, 5}
Set A is shown by Venn diagram.
1. 2.
.3 .4
.5 A
B = {x | -10 ≤ x ≤ 0, x is an integer}
Venn diagram given alongside represents the set B.
0 -1 -2 -3 -4
-5 -6 -7 -9
10 B
subset :
If A and B are two given sets and every
element of set B is also an element of set A then
B is a subset of A which is symbolically
written as B ⊆ A. It is read as ' B is a subset of A'
or ' B subset A '.
Ex (1) A = { 1, 2, 3, 4, 5, 6, 7, 8} B = {2, 4, 6, 8}
Every element of set B is also an element of
set A.
∴ B ⊆A.
This can be represented by Venn diagram as shown abov
e.
1 A 3
8 2
5 4
6 B
7
Ex (2)
N = set of natural numbers. I = set of integers.
Here N ⊆ I. because all natural numbers are integers..
Ex (3) P = { x | x is square root of 25} and
S = { y | y ∈ I, -5 ≤ y ≤ 5}
Let’s write set P as P = {-5, 5}
Let’s write set S as S = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
Here every element of set P is also an element of set S.
∴ P ⊆ S
Remember this
!
(i) Every set is a subset of itself. i.e. A ⊆ A
(ii) Empty set is a subset of every set i.e. ∅ ⊆ A
(iii) If A = B then A ⊆ B and B ⊆ A
(iv) If A ⊆ B and B ⊆ A then A = B
Ex. If A = { 1, 3, 4, 7, 8} then write all possible subsets
of A.
Universal set
Think of a bigger set which will accommodate all the
given sets under consideration which in general is known as Universal set. So
that the sets under consideration are the subsets of this Universal set.
Ex (1) Suppose we want to study the students
in class 9 who frequently remained absent.
Then we have to think of all the students of class 9 who
are in the school. So all the students in a school or the students of all the
divisions of class 9 in the school is the Universal set.
Let us see the another example.
Ex (2) A cricket team of 15 students is to be
selected from a school. Here all the students from school who play cricket is
the Universal set. A team of 15 cricket players is a subset of that Universal
set.
Generally, the universal set is denoted
by ‘U’ and in Venn diagram it
is
represented by a rectangle.
Complement of a set
Suppose U is an universal set. If B⊆U, then the set of all elements in U,
which are not in set B is called the complement of B.
It is denoted by B'
B' is defined as follows.
B' = {x | x ∈ U, and x ∉ B}
Ex (1) U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 1 3
A = {2,
4, 6, 8, 10}
5 7 2
4 6
∴ A' = {1, 3, 5, 7, 9}
9 8 10
Ex (2) Suppose U = { 1, 3, 9, 11, 13, 18, 19}
B = {3, 9, 11, 13}
1 18
∴ B' = {1, 18, 19}
19 3 9
11 13
Find (B')' and draw the inference.
(B')' is the set of elements which are not in B' but in U.
is (B')' = B ?
Understand this concept with the help of Venn diagram.
Remember this !
Complement of a complement is the given set itself.
Properties of complement of a set.
(i) No elements are common in A and A'.
(ii) A ⊆ U and A' ⊆ U
(iii) Complement of set U is empty set. U' = ∅