Given the Boolean equation Y = ABC + BC, a. List the complete truth table for Y. b. Draw the Venn diagram for the Boolean variables A, B, and C. Shade the area corresponding to Y. Use the Boolean Postulates and Theorems proven so far to prove the following theorems. Show and justify each step by citing the theorem or postulate used (refer to table 2.2 of your textbook). You must not cite a theorem as proof of itself. a. A middot 0 = 0 b. A(A + B) = AB c. (A + B)(A + B) = A d. AB + ABC = AB + AC e. (A +B)(A + C)(B + C) = (A + B) (A + C) Use the Boolean Postulates and Theorems proven so far to simplify the following expressions as far as possible. Show and justify each step by citing the theorem or postulate used (refer to table 2.2 of your textbook). a. f(w, x, y, z) = x + xyz + xyz = wx + wx + xy b. f(a, b, c, d, e) = (ab + c + d) (c + d)(c + d + e) c. f(x, y, z) = yz (z + zx) + (x + z) (xy + xz)