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Suppose A's possible values are ON or OFF.

Suppose I represent it as: if A ON then feature f=1 else f=0

Or, suppose I represent it with 2 features, where:

-if A is ON then f1=1 and f2=0

-if A is OFF then f1=0 and f2=1

How this kind of representation affects neural networks?

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It will have very little effect

The answer most will give is that it will have no effect, but adding one more feature will decrease the ratio of records to features so will slightly increase the bias and will hence make your model slightly less accurate. Unless, of course, you have overfit your model , in which case it will make your model slightly more accurate (a good data scientist would never do this because they understand the importance of cross-validation :-).

If you normalize your data and then attempt some sort of dimensionality reduction, your algorithm will immediately eliminate the feature that you added since it is perfectly negatively (linearly) correlated with the first feature. In this case it will have no effect.

Please also consider the following:

I always see big red flags when someone asks a very fundamental data science question with the words neural network included. Neural networks are very powerful and receive a great deal of attention in the media and on Kaggle, but they take more data to train, are difficult to configure, and require much more computing power. If you are just starting out, I suggest getting a foundation in linear regression, logistic regression, clustering, SVMs, decision trees, random forests, and naive Bayes before delving into artificial neural networks. Just some food for thought.

Hope this helps!

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