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Inés Aréizaga Esteva (Spain), 25 years old Education: Chemical Engineer NNE Pharmaplan is the world’s leading engineering and consultancy company focused entirely on the pharma and biotech industries. We employ more than 1500 people worldwide and offer global reach and local knowledge along with our all-encompassing list of services. com 30 Click on the ad to read more An Introduction to Abstract Algebra Preliminaries Example: be defined by f (x) = x2 Let f :→ Z+ • If A = {−3, −2, −1, 0, 1, 2, 3} then f (A) = {0, 1, 4, 9} .

As before for a uniqueness proof, we will begin by assuming that the identity is not unique. Suppose by way of contradiction that e, f are both identity elements of G and that e = f . However, e = f · e since f is an identity and f · e = f since e is an identity. Combining these equalities we see e = f · e = f . Our supposition is false and the identity element of a group is unique. ☐ Note: The hypothesis that we are working with a group is actually a little more than we need. Nothing in the above proof required any part of the definition of a group.

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