Exercise 1 [4 x 5% = 20%] Prove the following using logic manipulations (quote at each step the applied law). 1. (X' +Y)(X+Y) =Y 2. X'Y' + XY + X'Y + XY' = 1 3. W'Y'Z' + WY' + XYZ+W'YZ' + W'XY +WX'Y' = W'Z' + WY' + XYZ 4. Y + X'Z + XY' = X + Y + Z Exercise 2 [5 x 6% = 30%] Using logic manipulations simplify (quote at each step the applied law) each of the following expressions. 1. AB'C'D + AB'CD 2. (AB' + D)(AB'+D+CF) 3. (AB + C')(A' + B' + C) 4. (A + DE' + F)(DE' + (A+F)') 5. (1 + A' + B +C)(AB' + A + A') Exercise 3 [ 2 x 10% = 20%] Given f(a,b,c) = ab + b'c 1. Using logic manipulations, express f as a minterm expansion (use m notation) 2. Using f and logic manipulations, express f' as a maxterm expansion (use M notation). Exercise 4 [ 4 x 7.5% = 30%] Given f = m(0,1, 6, 7, 9, 11, 13, 15). 1. Using a K-map, find the minimum sum-of-products expression for f. 2. Using f and logic manipulations, find the minimum product-of-sums expression for f'. 3. Using a K-map, find the minimum sum-of-products expression for f'. 4. Using f'and logic manipulations, find the minimum product-of-sums expression for f