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Subset of a Set

     
A set A is said to be a subset of the set B is each element of the set A is also the element of the set B. The symbol used is ''  i.e. A  B <=> (x ε A => x ε B). 


Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A
 B.
e.g. If A = {a, b, c, d} and B = {b, c, d} then B
A or equivalently A B (i.e. A is a super set of B). 
 

Equality of Two Sets

Sets A and B are said to be equal if A  B and B  A and we write A = B. 

Universal Set

         

As the name implies, it is a set with collection of all the elements and is usually denoted by U. e.g. set of real numbers R is a universal set whereas a set A = [x : x < 3} is not a universal set as it does not contain the set of real numbers x > 3. Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to that set. e.g. If A = {x : x < 3 then
(or Ac) = complimentary set of A = {x : x > 3}. Hence we can say that A U = U i.e. Union of a set and its complimentary is always the Universal set and A  = f i.e. intersection of the set and its complimentary is always a void set. Some of the useful properties of operation on sets are as follows:


 

 

 

 

 

 

 

 

Example:

If A = {a, b, c} and B = {b, c, d} then evaluate A υ B, A ∩ B, A - B and B - A. 

Solution: 
A U B = {x : x ε A or x ε B} = {a, b, c, d} 
A ∩ B = {x : x ε A or x ε B} = {b, c} 
A - B = {x : x ε A and x  B} = {a} 
B - A = {x : x ε B and x  A} = {d} 
 

 





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