digital system hw

this hw is due in 5 hours, 

In all of the following problems use an appropriately sized K-map to perform the indicated minimization.

1) For each of the following function expressions generate a minimal sum of products expression:

  a) F (x,y,z) = 

   b) F (w,x,y,z) = 

2) For each of the following function expressions generate a minimal product of sums expression:

   a) F (x,y,z) = 

   b) F (w,x,y,z) = 

3) Design a circuit that performs modulo 4 multiplication. The circuit has two 2-bit numbers (A₁A₀ and B₁B₀) as the inputs. It produces a 2-bit number (C₁C₀) as the result. The output value is produced by multiplying the values of the two inputs and then computing the modulo 4 remainder. So if A was 2 and B was 3, their product is 6 and the remainder after dividing by 4 is 2. Develop minimized sum of product expressions for C₁ and C₀.

4) Design a circuit that performs some shifting and complementing of an input value. Assume there is a 3-bit input (A₂A₁A₀) and there are two control bits (C₁C₀) that define the operation to perform according to the following: When C₁C₀ = 00 shift the input bits to the right so they become A₀A₂A₁; When C₁C₀ = 01 shift the input bits to the left so they become A₁A₀A₂; When C₁C₀ = 10 shift the input bits to the right and complement them so they become A₀'A₂'A₁'; When C₁C₀ = 11 shift the input bits to the left and complement them so they become A₁'A₀'A₂'. Develop minimized sum of product expressions for the three output bits that result from shifting and complementing the input bits according to the control values.

5) Design a circuit that compares two 3-bit numbers ( A₂A₁A₀ and B₂B₁B₀ ) and produces a 2-bit result (C₁C₀) indicating the relative magnitudes of the two values. If the value of A is the same as the value of B, the output should be 11. If the value of A is greater than the value of B, the output should be 10. If the value of A is less than the value of B, the output should be 01. The output combination of 00 will never occur. Develop minimized sum of product expressions for the two output bits that result from performing the magnitude comparison.