Mapping/Squaring/Injective and surjective/Example

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The mapping

is neither injective nor surjective. It is not injective, because the different numbers and are both sent to . It is not surjective, because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

is injective, but not surjective. The injectivity can be seen as follows: If , then one number is larger, say

But then also , and in particular . The mapping

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

is injective and surjective.