RELATIONS

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Relations
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product         A Х B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A Х B.
  • The second element is called the image of the first element.
  • The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.
  • The set of all second elements in a relation R from a set A to a set B is called the range of the relation R.
  • The whole set B is called the codomain of the relation R. 
  • Range ⊆ codomain.
Example
    RELATIONS
  • Let A = { 2, 3, 4, 5 } and B = { 3, 6, 7, 10 }. A relation R from set A to the set B as follows: R = { ( x, y ) : x divides y, where x ∈ A and y ∈ B }.

We find that 2 divides 6 and 10, 3 divides 3 and 6, 5 divides 10 and there is no number in B that can be divided by 4.

Thus ( 2, 6 ) ∈ R, ( 2, 10 ) ∈ R, ( 3, 3 ) ∈ R, ( 3, 6 ) ∈ R, and ( 5, 10 ) ∈ R.
Therefore R = { ( 2, 6 ), ( 2, 10 ), ( 3, 3 ), ( 3, 6 ), ( 5, 10 ) }.
  • Domain of R = { 2, 3, 5 }
  • Range of R = { 3, 6, 10 }
  • Codomain of R = { 3, 6, 7, 10 }
  • Also R-1 = { ( 6, 2 ), ( 10, 2 ), ( 3, 3 ), ( 6, 3 ), ( 10, 5 ) }.
A relation may be represented algebraically either by the Roster method or by the Set-builder method.
An arrow diagram is a visual representation of a relation.

  • The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A Х B. If n(A) = p and n(B) = q, then n(A Х B) = pq and the total number of relations is 2pq. 
Inverse relations
Let A and B be two sets and R be a relation from the set A to the set B. Then the inverse of R, denoted by R-1 is a relation from the set B to the set A and is defined by 
  • R-1 = { ( b, a ) : ( a, b ) ∈ R }
  • Dmoain of R = Range of R-1  
  • Range of R = Domain of R-1  

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