I assume you are talking about machine learning.
A training set is the set of data you use to train a machine learning algorithm – think of it as studying for a test by using old tests. A test set is what you use to confirm that the algorithm has learned to apply to real situations rather than just overfitting to the test set – to continue the test analogy, the real test has to be different from the old tests you use to study, so that you actually have to learn the material rather than just memorize the answer.
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