Which of the following 2 input Boolean logic functions is linearly inseparable…
2016
Which of the following 2 input Boolean logic functions is linearly inseparable ?
(a) AND (b) OR
(c) NOR (d) XOR
(e) NOT XOR
- A.
(a) and (b)
- B.
(b) and (c)
- C.
(c), (d) and (e)
- D.
(d) and (e)
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Correct answer: D
Answer: XOR and NOT XOR (XNOR) are the linearly inseparable functions.
Reasoning:
Map the two binary inputs to points in the plane: (0,0), (0,1), (1,0), (1,1).
For XOR, outputs 1 at (0,1) and (1,0) while outputs 0 at (0,0) and (1,1). The 1s and 0s sit on opposite corners of the square, so no single straight line can separate the classes.
For XNOR (NOT XOR), outputs 1 at (0,0) and (1,1) with 0s at the other two corners; this is the complement of XOR and is also not linearly separable for the same geometric reason.
By contrast, functions such as AND, OR, and NOR are linearly separable. Examples of simple linear separators:
AND: a boundary like x1 + x2 > 1.5 separates the single positive point (1,1).
OR: a boundary like x1 + x2 > 0.5 separates the three positive points (1,0),(0,1),(1,1).
NOR: as the complement of OR, NOR is separable as well (for example with weights -1,-1 and bias 0.5 to single out (0,0)).