Operations
on sets
Intersection
of two sets:
Suppose A
and B are two sets. The set of all common elements of A and B is called the
intersection of set A and B.
It is denoted as AበB and read as A intersection B.
∴ A በB = {x | x ∈ A and x ∈ B} .
Ex (1) A = { 1, 3, 5, 7} , B = {
2, 3, 6, 8} The element
3 is common in set A and B. ∴ A በB = {3}
Ex (2) A = {1, 3, 9, 11, 13} , B = {1,
9, 11}
The elements
1, 9, 11 are common in set A and B.
∴ A በB = {1, 9, 11} But B = {1, 9, 11}
∴ A በB = B
Here set B
is the subset of A.
∴ If B ⊆ A then A በB = B, similarly, if B በA= B , then B ⊆ A.
Remember this !
Properties of Intersection of sets
(1) A በB = B በA (2) If A ⊆ B then A በB = A
(3) If A በB = B then B ⊆ A (4) A በB ⊆ A and A በB ⊆ B
(5) A በA’= ∅ (6) A በA= A (7) A በ∅ = ∅
Disjoint
sets
Let, A = {
1, 3, 5, 9}
and B = {2,
4, 8} are given.
Confirm that
not a single element is common in set
A and B.
These sets are completely different from each other.
So the set A
and B are disjoint sets. Observe its Venn diagram.
Union
of two sets
Let A and B
be two given sets. Then the set of all elements of set A and B is called the Union
of two sets.
It is written as A ∪ B and read as 'A union B'.
A ∪ B = {x | x ∈ A or x ∈ B}
Ex (1) A = {-1, -3, -5, 0}
B = {0, 3,
5}
A ∪ B = {-3, -5, 0, -1, 3, 5}
Note that, A
∪
B = B ∪ A
Observe the Venn diagram and write the following sets
(iv) A ∪ B (v) A በB
(vi) A' (vii) B' (viii)(A ∪ B)'
(ix) (A በB)'
Solution : We Have, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 8, 10}
Now ,A U B ={1, 2, 3, 4, 5, 6, 7, 8, 10} , A በB = {8, 10}
A’ = {1, 3,
5, 7, 9, 11, 12} , B’ = {2, 4, 6, 9, 11, 12}
(A U B)’ ={9, 11, 12}
, (A በB)’ = {1, 2, 3, 4, 5,
6, 7, 9, 11, 12}
Ex (3) A
= {1, 2, 3, 4, 5} , B =
{2, 3}
Let us draw its Venn diagram.
A U B = {1, 2, 3, 4,
5} Observe that set A
and A U B have the same elements. Hence, if B ⊆ A then A U B = A
Remember
this !
Properties
of Union of sets
(1) A U B =
B U A (2) If A ⊆ B then A U B = B
(3) A ⊆ A U B, B ⊆ A U B (4) A U A’= U
(5) A U A= A
(6) A U ∅= A
Number of
elements in Union and Intersection of
sets:
Let us
consider the set A and set B as given above,
n (A) + n
(B) = 5 + 6 = 11 ----(I)
A U B= {3,
6, 9, 12, 15, 18, 24, 30, 36} ∴ n (A U B) = 9--------(II)
To find A በB means to find common elements of set A and
set B.
A በB = {6, 12} ∴ n (A በB) = 2--------(III)
In n (A) and
n (B) elements in A በB are counted twice.
∴ n (A) + n (B) - n (A በB ) = 5 + 6 - 2 = 9
and n (A U B ) = 9
From
equations (I), (II) and (III), we can write it as follows
∴
n (A U B ) = n (A) + n (B) - n (A በB)
Verify the above rule for the given Venn diagram.
n (A) = ,
n (B) = A 24
n (A U B )= , n (A በB)= 9 3 6 18 B
∴ n (A U B ) = n (A) + n (B) - n (A በB) 15 12 30
36
Remember this !
n (A U B ) =
n (A) + n (B) - n (A በB)
means n (A)
+ n (B) = n (A U B ) + n (A በB)