A mapping is Surjective if and only if the codomain is equal to the range.
A mapping is Injective if every distinct element of the domain leads to or gives a distinct element in the co-domain.
A bijective function is a mapping that contains both properties of injective (one-to-one) and surjective (onto).
These properties build to form other identities and tools for graphing in computer programming, and image definitions in computers. Last semester I was able to write a python program that opened a file of a picture of text, and converted the picture into a character on the computer using the mapping relationship properties and linear algebra to define vectors and matrices. By taking a linear transformation on the data sets, and isolating the eigenvalues of the matrix which represents the image, I was able to generate code that enabled the image to be captured as text.
I am working on developing an application that will allow you to take a picture of grocery receipts and automatically balance your checkbook. I plan to convert this application into a tracking measure to also tabulate the rounding error that grocery stores take on individual transactions, and use it to ask for a reimbursement from stores that I frequent over the year!
I see how valuable the properties of injective, and surjective mappings are as a foundation of a lot of different mathematical tools! I hope to continue to learn about these principles and grow in understanding so that I can continue to explore the possibilities of application of math in my daily life!
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