Definitions [6]
Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e., P × Q = { (p,q) : p ∈ P, q ∈ Q } If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = Ø
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × 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. Note that range ⊂ codomain.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.
A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.
Define an equivalence relation ?
A relation R on set A is said to be an equivalence relation if
(i) it is reflexive,
(ii) it is symmetric and
(iii) it is transitive.
Relation R on set A satisfying all the above three properties is an equivalence relation.
Define a symmetric relation ?
A relation R on a set A is said to be symmetric if
(a, b) ∈ R
⇒ (b, a) ∈ R for all a b ∈ A
i.e. aRb ⇒ bRa for all a , b ∈ A
Define a transitive relation ?
A relation R on a set A is said to be transitive if
(a, b) ∈ R and (b, c) ∈ R
⇒ (a, c) ∈ R for all a, b , c ∈ R
i.e. aRb and bRc
⇒ aRc for all a, b, c ∈ R
