Sets

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This chapter deals with some basic definitions and operations involving sets. These are summarised below:

• A set is a well-defined collection of objects.
 
• A set which does not contain any element is called empty set.
 
• A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.
 
• Two sets A and B are said to be equal if they have exactly the same elements.
 
• A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R.
 
• A power set of a set A is collection of all subsets of A. It is denoted by P(A).
 
• The union of two sets A and B is the set of all those elements which are either in A or in B.
 
• The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B. 
• The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A. 
• For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′
 
• If A and B are finite sets such that A ∩ B = φ, then  
n (A ∪ B) = n (A) + n (B).
If A ∩ B ≠ φ, then
n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

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Relation A relation R from a non-empty set A to a non empty set B is a subset of the Cartesianproduct A × B.  The set of all first elements of the ordered pairs in a relation R from aset A to a set B is called the domain of the relation R.  The set of all second elements ina relation R from a set A to a set B is called the range of the relation R.  The whole setB is called the codomain of the relation R.  Note that range is always a subset ofcodomain. A relation R on set is the subset of A×A. Therefore,  R ⊂ A × A. Types of Relations Extreme Relation 1. Empty  A relation R in a set A is called empty relation, if no element of A is related to anyelement of A, i.e., R = φ ⊂ A × A. 2. Universal  A relation R in a set A is called universal  3. Reflexive  A relation R in A is said to be reflexive if aRa for all a∈A 4. Symmetric  R is symmetric ifaRb ⇒ bRa, ∀ a, b ∈ A 5. Transitive it is said to be transitive if aRb and bRc ⇒ aRc∀ a, b, c ∈ A ...
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