SET AND VENN DIAGRAM
Sets : A collection of well defined objects is called a set. e.g. set of chairs, tables, tall boys, fat girls etc.
The objects are known as elements of the set. There are two ways of representing a set. viz.
- Roster Method : A set is represented by putting its elements within a bracket( ).
- Set Builder Form : In set builder form the objects (elements) of a set are written as per some rule or
Equality of sets : Two sets A and B are said to be equal if every element of A is a member of B and every element of B is also a member of A.
In other words we call two sets equal if they have some identical elements. e.g. A = (1, 2, 3,) and B = (3, 1, 2)
One to one (1-1) correspondence : Two sets have one to one correspondence if each element of the first set can be assigned a unique. object of the second and vica- versa.
If two sets have equal number of elements, they have one to one correspon- dence.
Equivalent sets : Two sets A and B are called equivalent sets if they have a one-to-one correspondence between the elements, i.e. if the number of elements in A are equal to the number of elements in B.
Two sets equivalence is denoted by the symbol ‘~’. Hence A ~ B is read as ‘A is equivalent’ to ‘B’.
Example : Let A = {2, 3} and B = {4, 5}.
Subset : If each element of A is a member of B, we call A as subset of B and express if A £ B and read it as ‘A is a subset of B’.
This can also be expressed as B ³ A (B is a super set of A).
e.g. The set of vowels (V) is a subset of alphabets (A) i.e. V Í A.
Proper Subset : If each element of A is a member of B, the set A is called the proper subset of B. It is written as AÌB (A is a proper subset of B)
This may also be written as BÉA. e.g. set of natural number is a proper subset of whole numbers i.e. NC W but W Ë N.
Universal Set : When all sets being considered are the subsets of a bigger set, the bigger set is known as universal set. The universal set is shown by È or X or x
e.g. È = (1, 2, 3, 5, 6)
A = (1, 2, 3)
B = (1, 5, 6)
Here both A and B Ì È, so È is a universal set.
Complement of a set : Consider a universal set È and A be any set then A Ì È, then complement of set A is shown by A1, as
A1 = { X : x e È, x Ë A}
If A = (1,2,3), È = (1, 2, 3, 4, 5) then A1 = (4, 5)
Overlapping sets : If a set A and set B have some common elements, then the
overlapping sets are A Ç B Ë j
e.g. set A = (1,2,3) B = (2,3,4)
here 2 and 3 are common elements in A and B, hence they are overlapping sets. Finite sets : Any set which has finite number of elements is known as finite set. Empty or Null set : A set without any element is called a null set. It is written as
j or { }
P (A)
Unit set : A set consisting of only one element is called unit set.
Power set : The set of a given set A is known as power set of A and is written as
If A = {a,b,c}
P (A) = {j}, {1}, {b}, {c}, {a,b}, {a,c,} {b,c} {a,b,c}
Disjoint sets : If A and B are two sets such that A Ç B = j, these two sets are
called as disjoint sets.
e.g. A = {1,3,5… }
B = {2,4,6,8. }
these two are disjoints sets
Properties of union of sets :
- A Ç B = B Ç A.
- (A Ç B) Ç C = A Ç (B Ç C)
- A Ç j = j
- A Ç U = A, where U is the universal
- A Ç A =
- If A Ç B = j, then A and B are disjoint
- If A Ç B = j, then A and B are overlapping
- If A Ì B, then A Ç B =
- If B Ì A, then A Ç B =
(2) The intersection of two sets A and B is the set whose elements are in A and B
both.
A Ç B = {x : x e A and x e B} Read A Ç B as A intersection B.
Definition in symbols : A – B = (x : x e A and x Ë B).
Example : If A = {1,2,3,7,11}.
B = {2,7,3,6}, then
A – B = {1,3,11} and B – A = {3, 6}.
Properties of difference of two sets :
- A – B = A Ç B’
- A – B = j, if and only if A Ì
- A – B = B – A, if and only if A Ç
- A – B = A, if and only if A Ç B = j
De Morgan’s laws :
- The complement of the intersection of two sets is equal to the union of their
complement i.e, (AÇB)’ = A’ È B’
- The complement of the union of two sets is equal to the intersection of their complements e.,
(A È B)’ = A’ Ç B’
suppose È is a universal set, so every element under consideration belongs to X
(1) (A Ç B)’ = {x : x Ï A Ç B}
= {x : x Î A’ or x Î B’}
= A’ = Ç B’.
(2) (A Ç B)’ = A’ È B’.
Some Important Formulae : For any three sets A, B, C.
(1) n(A È B) = n(A) + n(B) – n(A Ç B)
(2) If A Ç B = j, then n(A È B) = n(A) + n(B)
(3) n (A – B) + n(A Ç B) = n(A)
(4) n (B – A) + n(A Ç B) = n(B)
(5) n(A È B) = n(A – B) + n(A Ç B) + n(B – A)
(6) n(A È B È C) = n(A) + n(B) + n(C) – n(A Ç B) – n(B Ç C) – n(C Ç A) + n(A Ç B Ç C)
