Which Boolean operation on two variables can be represented by a single…

Answer: AND, OR and NOR (XOR cannot be represented by a single-layer perceptron).

Reason: A single-layer perceptron implements a linear decision boundary of the form w1·x1 + w2·x2 + b ≥ 0. Any Boolean function whose true and false outputs are separable by a single line in the input space (i.e., linearly separable) can be implemented by such a perceptron. XOR is not linearly separable and therefore requires at least a two-layer network.

• AND — linearly separable. Example weights: w1 = 1, w2 = 1, bias b = -1.5 (output 1 only when both inputs are 1).

• OR — linearly separable. Example weights: w1 = 1, w2 = 1, bias b = -0.5 (output 1 when at least one input is 1).

• NOR — negation of OR and also linearly separable. Example weights: w1 = -1, w2 = -1, bias b = 0.5 (output 1 only when both inputs are 0).

• XOR — not linearly separable; true cases (inputs (1,0) and (0,1)) cannot be separated from false cases with a single linear boundary, so a single-layer perceptron cannot implement XOR.

Conclusion: AND, OR and NOR can be represented by a single perceptron; XOR cannot.