Law of Algebra of sets:
a) Impotent law- A A=A and A
A = A
b) Identity Law – A = A and A
u=A
c) Commutative law- A B = B
A and A
B = B
A
d) Associative law- (A B)
C = A
(B
C)
e) Distributive law- A (B
C) = (A
B)∩(A
C)
– A (B
C)= (A
B)
(A
C)
Some Use Full Theorem
If A and B are any two sets then
a) A-B = A
b) B-A=B
c) A-B=A A
B =
d) (A-B) B = A
B
e) (A-B) B =
f) A B
g) (A-B) (B-A) = (A
B) – (A
B)
Some Important Result on Number of Elements In SETS
If A,B and C are finite sets and u be the universal set then.
1. n(A B) = n(A) + n(B) – n(A
B)
2. n(A B) = n(A) + n(B) + n(B)
A,B are adjoint non vaid sets.
3. n(A-B) = n(A) – n(A B)
i.e n(A-B) + n(A B) = n(A)
4. n(AB) = number of elements which belong to exactly one of A or B
= n(A-B)(B-A)
= n(A-B) +n(B-A) ……………. [(A-B) and (B-A) are disjoint]
= n(A)- n(A B) + n(B) – n(A
B)
= n(A) + n(B) – 2n(A B)
5. n(A B
C) =
n(A) + n(B) + n(c) – n(A∩B) –n(B C)-n(A
C) + n(A
B
C)
6. number of elements in exactly two of the sets A,B,C
= n(A B) + n(B
C) + n(C
A)- 3n(A
B
C)
7. number of elements in exactly on cot the sets A,B,C
= n(A) + n(B) + n(c) – 2n(A B) –2n(B
C)-2n(A
C) + 3n(A
B
C)
8. n( ) = n((A∩B)’) = n(u) – n(A
B)
Certesian Product Of Sets
Ordered Pair– an ordered pair consists of two objects or elements in a given fixed order
If A&B are any two sets then by an order pair of elements are (a,b) Where a A and b
B
The position of a paint in two dimensional plane intercession coordinate is represented by an order pair (-1,5) Where x R and Y
r
CARTESIAN PRODUCT OF SETS: if A and B are two non empty sets the set of all ordered pairs (a,b) such as a a A and b
B is called Cartesian product of the set A and B and is denoted by AB
A×B = {a,b}: a A and b
B}
Example if A= (x,y) and B= (2,3,4) find A×B, B×A, A×A
A×B = {(x,z), (x,3), (x,4), (y,z), (y,3), (y,4)}
B×A= {(z,x), (z,y), (3,x), (3,y), (4,x), (4,y)}
A×A= {(x,x), (x,y), (y,x), (y,y)}
As above (A×B) (B×A) =
Graphical Representation Of Cartesian Product Of Sets
If A = {2,3,4}
B = {3,4}
A×B = {(2,3) (2,4) (3,3) (3,4), (4,3), (4,4)}
To represent A×B graphical, draw to line perpendicular to each X & Y axis and then draw these pairs.
As above n(A×B) = n(A), n(B) = 2×3 = 6 Pair
A×B = A =
, B =
A×A×A= {(a,b,c) : a,b,c A}
(a,b,c) is called an ordered triplet
(i) A× (BUC) = (A×B)U (A×C)
(ii) A×(B∩C) = (A×B)∩ (A×C)
(iii) A×(B-C) = (A×B)- (A×C)
(iv) A×B = B×A A=B
(v) A≤B then A×A ≤ (A×B) (B×A)
(vi) A≤ B then A×C ≤ B×C
(vii) A≤B and C≤D than A×C ≤ B × D
(viii) For any set Four A,B,C,D
(A×B)∩ (C×D) = (A∩C) × (B∩D)
For any set A and B
(A×B)∩ (B×A) = (A∩B) × (B∩A)
(ix) For any three set A,B,C
A × (B’UC’) = (A×B) ∩(A×C)
A× (B’∩C’) = (A×B)U (A×C)
(x) If any two non empty sets have n elements in common. Then A×B and B×A have elements in common.
(xi) If A is non empty set and A×B = A×C
Then B=C