1. Prove the following theorems algebraically: (a) X(X + Y) = XY (b) X+XY = X (c) XY+XY' = X (d) (A + B)(A+B' )= A 2. Obtain the sum of products for the following (a) (A+B)(C+B)(D' +BXACD' +E) (b) (A' +B+C' A' + C + D (B' +D') 3. Obtain product of sums for the following, (b) WX + WY' X + ZYX (C) A' BC + EF + DEF' 4. Draw a circuit that uses only one AND gate and one OR gate to realize each of the following functions: (2.7) (a) (A + B + C + DXA + B + C + EXA+B+C+F) (b) WXYZ + VXYZ +UXYZ 5. For each of the following circuits, find the output and design a simpler circuit that has the same output. (Hint: Find the circuit output by first finding the output of each gate, going from left to right, and simplifying as you go.) TDID D- PDABD DD