If is an infinite set, then there is a bijection from S onto a proper subset of

Question

If is an infinite set, then there is a bijection from S onto a proper subset of S. Use two of the sets or any of their subsets or supersets of N,Q,R,Z to illustrate this fact. Define the bijection between them.

Explanation / Answer

Take N--Set of natural numbers{1,2,3,4.................} E--Set of even numbers{2,4,6,8....................} E is the proper subset of N We define mapping f:N----->E by f(x)=2x Now we will show that this function is one-one and onto First we will show one-one Let f(x1)=f(x2) =>2x1=2x2 =>x1=x2 hence f is one one Now we will show function is onto let t be any even number belonging to E since t is even we can write it as t=2s for some "s" belonging to Natural number. hence f(s)=2s=t hence for every t belonging to E there exist s belonging to N Hence f is both one one and onto hence bijective.

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