when trees fall...

In 1969, Marvin Minsky and Seymour Papert published Perceptrons: An Introduction to Computational Geometry. In it, they showed that a single-layer perceptron cannot compute the XOR function. The main argument relies on linear separability: Perceptrons are linear classifiers, which essentially means drawing a line to separate input that would result in 1 versus 0. You can do it in the OR and AND case, but not XOR. Of course, we're way past that now, neural networks with one hidden layer can solve that problem. The solution in essence is analogous to composing AND, OR, and NOT gates, which can be represented by single-layer networks, to form the required function. Depth is important for certain functions.