The perceptron uses a hyperplane to separate the positive and negative classes. A simple example of a classification problem that is linearly inseparable is the logical exclusive disjunction, or XOR. The output of XOR is 1 when one of its inputs is equal to 1 and the other is equal to 0. Otherwise, the output is 0. The inputs and outputs of XOR are plotted in two dimensions in the following graph. When XOR outputs 1, the instance is marked with a circle; when XOR outputs 0, the instance is marked with a diamond:
It is impossible to separate the circles from the diamonds using a single straight line. Suppose that the instances are pegs on a board. If you were to stretch a rubber band around both of the positive instances, and stretch a second rubber band around both of the negative instances, the bands would intersect in the middle of the board...
