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 ->

Explanation

(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.