Union of sets in O(N)


Hi,

Suppose A,B,C.....are sets which contains real numbers between 0 and 1,

(U=Union)
I need to create a set S= AUBUCU...... in O(N)

It is also known that for every x,y belong to S

MIN(ABS(X-Y))>1/m (x!=y)

Where 1/m=O(N)


I am planning to use hash table to solve it but I wonder can I prove
from the above that a hash function without hash collision exist?

Thanks in advance.

.

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