InterviewSolution
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InterviewSolution
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Check the injectivity and surjectivity of the following functions:(i) `f : N ->N`given by `f(x)=x^2`(ii) `f : Z-> Z`given by `f(x)=x^2`(iii) `f : R ->R`given by `f(x)=x^2`(iv) `f : N-> N`given by `f(x)=x^3`(v) `f : Z -> |
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Answer» (i) `f:N to N and f(x) = x^(2)` Let `x, y in N and f(x) -f(y)` `rArr " " x(2) =y^(2)` `rArr " " x=y ( because x,y in N)` `therefore f` is one-one. Let `f(x) = y` where `y in N` `rArr x ^(2) =y ` `rArr x = pm sqrty notin N ` if `y =2` `therefore f` in not onto. Therefore, `f` is one-one but not onto. (ii) `f: Z to Z and f(x) = x^(2)` Let `x, y in Z and f(x) = f(y)` `rArr x ^(2) = y^(2) rArr x = pm y ` `therefore f `is not one-one. Again, let `f(x) = y` where `y in Z` `rArr x ^(2) =y` `rArr x = pm sqrty notin Z ` If ` y = 2 ` ` therefore f` is not onto. Therefore, `f` is neither one-one nor onto. (iii) `f : R to R and f(x) = x^(2)` Let `x, y in R and f(x) = f(y)` `rArr x^(2) = y^(2) rArr x = pm y ` `therefore f ` is not one-one Again, let `f(x) = y ` where `y in R` `rArr x^(2) = y` `rArr x = pm sqrty notin R` if ` y = -2` ` therefore f` is not onto. Therefore, `f` is neither one-one nor onto. (iv) `f: N to N and f(x) = x^(3)` Let `x , y in N and f(x) = f(y)` `rArr " "x ^(3) = y^(3) rArr x =y ` `therefore f` is one-one. Let `" " f(x) =y " " ` where `y in N ` `rArr x ^(3) = y ` ` rArr " "x = y ^(1//3) notin N ` if ` y = -2 ` `therefore f ` is not onto. Therefore, `f` is one-one but not onto. (v) `f: Z toZ and f(x) = x^(3)` Let ` x , y in Z and f(x) = f(y)` `rArr x ^(3) = y^(3) rArr x =y` `therefore f` is one-one. Again let `f(x) -y ` where `y in Z` `rArr x ^(3) = y` `rArr x = y^(1//3) notin Z ` if `y= 2` `therefore f ` is not onto. Therefore, `f` is one-one but not onto. |
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