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show that the function `(i) f : N to N : f (x) =x^(2)` is one-one into (ii) `f : Z to Z : f (x) =x^(2)` is many -one into . |
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Answer» Let x,y `in` N and f (x) = f(y) `rArr x^(2)=y^(2)` `rArr x^(2)-y^(2)` = 0 `rArr (x-y)(x+y) = 0` `rArr x-y =0` (`because x + y ne 0`) `rArr` x=y Therefore, f is one one. Again , let f (x) = y, where y `in` N (co-domain) `rArr x^(2) = y` `rArrx=+-sqrt(y)in N is y -3 in N` Therefore , an element 3 belongs to co-domain such that it has no pre-image in domain N. `:.` f is into. Therefore, f is one-one into. |
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