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Question: a boolean function fix2 is called a threshold function if...

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A Boolean function fix2) is called a threshold function if there are n+1 real numbers wi,w2...wn and T such that :. (xi, x2,. . .,น.) = 1 if and only if wixi + w2x2+ . . . waxoZ Tandrri,n, . . . №) = 0 otherwise, where the inequalities are numerical expressions involving plus and times For example, the AND function/x1,x2)-xin is a threshold function, which follows from setting wi = 0.5, w2 = 0.5, and T = 1.0 as follows 1S C wixi + w2x2 = 0.0 Tfor (x1,x2)-(0,0) =>/(0,0) = 0 w1X1 + wm = 0.5c Tfor (x1,x2) = (0,1) =>/(0,1) = 0 wixi + w2x2 = 0.5c Ttor (x1,x2)-(1,0) =>/(1,0) = 0 wïx1 + w2X2 = 1.02 T for (x1,x2)-(1,1) =>/(1,1) = 1 There are many other possible choices of wis and T that define AND in threshold function form. Threshold functions are of interest because they model (in a very simplified way) the behavior of a biological neuron (a) [5 points] Prove that the two-variable OR function is also a threshold function. (b) [10 points] Prove that the two-variable EXCLUSIVE-OR function is not a threshold function

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