One-to-One/Onto Functions Here are the definitions: is one-to-one (injective) if maps every element of to a unique element in . In other words no element of are mapped to by two or more elements of . . is onto (surjective)if every element of other words, nothing is left out. . In . is mapped to by some element of is one-to-one onto (bijective) if it is both one-to-one and onto. In this case the map is also called a one-to-one correspondence. Example-1 Classify the following functions onto. between natural numbers as one-to-one and One-to-One? Onto? Yes No Yes No No Yes Yes Yes . It helps to visualize the mapping for each function to understand the answers. Reasons is not onto because it does not have any element instance. is not onto because no element is not one-to-one since such that such that , for , for instance. . Example-2 Prove that the function is one-to-one. Proof: We wish to prove that whenever then for two numbers . Therefore, . Splitting cases on , we have For , , therefore for this case. . Let us assume that . Which means that For , we have . Therefore, it follows that for both cases. Example-3 Prove that the function is onto. Proof Given any , we observe that are mapped onto. is such that . Therefore, all Claim-1 The composition of any two one-to-one functions is itself one-to-one. Proof Let and be both one-to-one. We wish to tshow that is also one-to-one. Assume that for two elements . Therefore . Since is itself one-to-one, it follows that Since is one to one and Therefore . it follows that can happen only if The reasoning above shows that . . is one-to-one. Claim-2 The composition of any two onto functions is itself onto. Proof Let and be onto functions. We will prove that Let be any element. Since is onto, we know that there exists Likewise, since is onto, there exists Combining, Thus, is onto. . is also onto. such that such that . .