1. Construct a network, N1, with O(n) weights, which implements a function from R to {0,1}n and satisfies the following property. For every x ∈ {0,1}n , if we feed the network with the real number 0. x1x2 . . . xn, then the output of the network will be x.

Hint: Denote α =0. x1x2 . . . xn and observe that 10kα−0.5 is at least 0.5 if xk =1 and is at most −0.3 if xk =−1.

2. Construct a network, N2, with O(n) weights, which implements a function from [n] to {0,1}n such that N2(i )=ei for all i . That is, upon receiving the input i, the network outputs the vector of all zeros except 1 at the i ’th neuron.

3. Let α1, . . .,αn be n real numbers such that every αi is of the form 0.a(i ) 1 a(i ) 2 . . .a(i ) n , with a(i ) j ∈ {0,1}. Construct a network, N3, with O(n) weights, which implements a function from [n] to R, and satisfies N2(i ) = αi for every i ∈ [n].