x-y ≠ y-z, so if relation R will have( x, y), it will not have( y, x), so it is not symmetric.
If x-y = 7, & y-z = 7, then x-z = 14, not 7.
Thus if relation has( x, y) &( y, z) elements, it will not have( x, z), so it is not transitive.
Equivalence Class
If R is an equivalence relation on set A, then it decomposes A into pair wise disjoint subsets. All elements of a subset are related to one another under equivalence R and no element of a subset is related to an element in any other subset.
A = A1 + A2 + A3 + A4 … An Subsets A1, A2, A3,… An etc are called Equivalence class.
The equivalence relation partitions the set A into mutually exclusive equivalence classes.
o o
o
Let N be set of all natural number. n, m are members of N
Let R be equivalence relation defined b / w n & m.( m & n leaves same remainder when divided by 5).
N = A1 + A2 + A3 + A4 + A5 o A1 = { n; n is ∈ N, n leaves remainder 0 on division by 5 } o A2 = { n; n is ∈ N, n leaves remainder 1 on division by 5 } o A3 = { n; n is ∈ N, n leaves remainder 2 on division by 5 } o A4 = { n; n is ∈ N, n leaves remainder 3 on division by 5 } o A5 = { n; n is ∈ N, n leaves remainder 4 on division by 5 }
Types of Functions o One-one( or injective) o Onto( or surjective), o One-one and onto( or bijective)
Function: one-one( or injective)
o
A function f: X → Y is defined to be one-one( or injective), if the images of distinct elements of X under f are distinct, i. e., for every x1, x2 ∈ X, f( x1) = f( x2) implies x1 = x2, otherwise many-one.