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
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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 )
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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 .