The set of all subsets of a non-empty set “A” is called power set of “A”. It is denoted by P(A).
Number of elements in the power set of A = 2 ^
(s(J))is a finite family of sets, then the union of all the sets is denoted as A, cup A_{2} cup A 3 cup A 3 cup……….. cup A n bigcup i = 1 ^ n A i A_{1}, A_{2}, A_{3} ,………..A s
If A_{1}, A_{2}, A_{3} An is a finite family of sets, then their intersections one after another is given by A_{1} cap A 2 cap A 3 cap……….. cap A n or bigcap k = 1 ^ n A kLet A, B and Che any three sets and ‘U’ be the universal set, theni) A cup A = A and A cap A = A(Idempotent laws)ii) A cup Phi = A and A cap U = A(Identity laws)iii) U cup Lambda = U(Law of universal set)iv) A cup B=B cup A and A cap B=B cap A(Commutative laws)V) ( A cup B) cup C=A cup(B cup C) and ( A cap B) cap C=A cap(B cap C)(Associative laws)vi) A cup(B cap C)=(A cup B) cap(A cup C) and A cap(B cup C)=(A cap B) cup(A cap C)(Distributive laws)vii) (A cup B)^ =A cap B deg * and (A cap B)^ =A cup B^( D e^ * Morgan’s laws)
If A, B and C are finite sets and ‘U’ be the finite universal set, then
i) n(A cup B)= n(A) + n(B) -n(A cap B)
ii) n(A cup B)= n(A) + n(B) where A,B are disjoint non-empty sets
iii) n(A – B) =n(A)-n(A cap B) i.e., n(A – B) +n(A cap B)=n(A)
iv) n(A*Lambda*B) = n(A – B) + n(B – A) =n(A)+n(B)-2n(A cap B)
v) n(A cup B cup C)= n(A) + n(B) + n(C) -n(A cap B)-n(B cap C)-n(C cap A)+n(A cap B cap C)
vi) Number of elements in exactly two of sets A, B and C =n(A cap B)+n(B cap C)+n(C cap A) – 3n (A cap B cap C)
vii) Number of elements in exactly one of sets A, B and C = n(A) + n(B) + n(C) -2n(A cap B)-2n(B cap C)-2n(A cap C)+3n(A cap B cap C)
Maximum of (n(A), n(B)} ≤ n(AB) ≤ n(A) + n(B)0n(AB)
Minimum of {n(A), n(B))
Given two non-empty sets A and B, the set of all ordered pairs (a,b) such that a <^ and b \notin B are called Cartesian product of sets A and B and is denoted by AB AB = \{(a, b) / a \in A, b \in B\}
If “A” and “B” are two non-empty sets, then n(AB) = n(A) * n(B) = n(BA)
For any three sets A, B, C
i) A*(B cup C)= (AB) cup(A* C)] A*(B cap C)= (AB) cap(A* C) Distibutive property of cartesian product operator A(B – C) = (AB) – (AC)
ii) f’ * A’ and two -empty sets, then AB =B* A Leftrightarrow A=B
iii) If A subseteq B then A* A subseteq (AB) cap(B* A)
iv) If A subseteq B then A* C subset BC for any set C.
v) IF A subseteq B and C \in D then A* C subseteq BD
vi) For any sets A, B, C,D, (AB) cap(C* D)=(A cap C)*(B cap D)
vii) For any sets A and B (AB) cap(B* A)=(A cap B)*(B cap A)
viii) Let A and B be two non-empty sets having n elements in common, then AB and BA have n³ elements in common.ix) Let ‘A’ be a non-empty set such that AB = AC then B = CA relation ‘R’ is said to be relation on A, R subseteq AAIf n(A) = m then number of Relations from A to B is 2 n(B) = nClassifications of RelationsReflexive RelationSymmetric RelationTransitive RelationA relation ‘R’ on a set ‘A’ is said to be Reflexive if(a, a) \in R for alla e AThe number of reflexive relations from A to A is 2 ^ (n ^ 2 * n) where ‘n’ is the number of elements in set A.A relation ‘R’ on a set ‘A’ is said to be symmetric relation iff (a,b) in R Rightarrow (b, a) \in R , a, b \in AThe number of symmetric relations on a set containing ‘n’ elements is 2 ^ ((n(n + 1))/2)A relation R on a set A is said to be anti-symmetric relation iff (a, b) \in Rand(b, a) in R Rightarrow a=b:a,b in AA relation R on a set A is said to be transitive relation, iff f(a, b) \in Rand(b, c) in R Rightarrow(a,c) in R:a,b,c in AA Relation which is Reflexive, symmetric and transitive is an equivalence RelatioThe number of equivalence relations on a set A containing n elements
A relation is an ordered relation if it is transitive but not equivalence. sum r = 1 to n 1/(r!) \ r^ prime prime -r C 1 (r-1)^ n +r C 2 (r-2)^ n -r C 3 (r-3)^ n +….\
A relation is a partial ordered relation if it is Reflexive, transitive and Anti-sym
R ^ – 1 = \{(b, a) / (a, b) \in R\} . (a,b) in R Leftrightarrow (b, a) \in R ^ – 1
Domain (R) = Range(R¹); Range of R = Domain (R-¹).R ^ – 1
= R then R is a symmetric relation.
A binary operation on set ‘A’ is a function f / A * A -> A
The number of binary operations on set ‘A’ is n ^ (n ^ 2) where n(A) = n