Q. 153.8( 4 Votes )

If A = {1, 2, 3},

Answer :

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.


So, is One – One function


a≠b


f(a)≠f(b) for all


f(a) = f(b)


a = b for all


Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.


So, is Surjection iff for each , there exists such that f(a) = b


Now, f : A A where A = {1, 2, 3} and its an Onto function


To Prove: – A is a One – One function


Let's assume f is not Onto function,


Then,


There must be two elements let it be 1 and 2 in Domain A = {1, 2, 3} whose images in co–domain A = {1, 2, 3} is same.


Also, Image of 3 under f can be only one element.


Therefore,


Range set can have at most two elements in co – domain A = {1, 2, 3}


f is not an onto function


Hence it contradicts


f must be One – One function


Hence Proved


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