Page 156 - Engineering Digital Design

PROBLEMS 127

3.27 Reduce the following expressions to their simplest form, but do not name the Boolean
laws used in each step.
(a) A + AfiC + A + C
(b) (acd + ad)(ad + cd)
(c) (WX + Y + W)(WX + Y + WX)
(d) (x + y)(x + z)(y + z)
(e) (A + BC)(AB + ABQ (Hint: Simplify under short complementation bars first.)
(f) a + b + a(b + be} + (b + c) • abed
(g) (A + B + C + D)(A + C + D)(A + B + D) (Hint: First use consensus.)
3.28 Reduce the following expressions to their simplest form and name the Boolean laws
used in each step.
(a) (a 0 b + b)(a + b)
(b) (XY) 0 (X + Y)
(c) x O y O (xy)
(d) [(X + Y) O (X + Z)] + X
(e) [(A + B) • C] 0 [A + B + AC] (Hint: Find a way to use Corollary II.)
3.29 Reduce the following expressions to their simplest form, but do not name the Boolean
laws used in each step.
(a) A + A 0 B + AB
(b) [S + [S 0 (ST)]}(H) =
(c) (X + 7) O (X 0 7)
(d) (a O ft) 0 (ah)
(e) (* + y)(* 0 y + }0
(f) [1 0 (I+Ol) + 1 O 0](H) = [?](L)

3.30 Use the laws of Boolean algebra, including the XOR laws, identities, and corollaries
given by Eqs. (3.17) through (3.33), to prove whether the following equations are true
(T) or false (F). Do not name the laws used.
(a) X O (X + 7) = XY
(b) ab(b + be) + bc + abed = be
(c) A 0 B 0 (AB) = AB
(d) X 0 (XT) = X + (X Q Y)
(e) [(AB)(A O B)](L) = AB(H)
(f) AXY + AXY + AY = (AX) 0 Y (Hint: First apply Corollary I.)

3.31 Use whatever conjugate gate forms are necessary to obtain a gate-minimum imple-
mentation of the following functions exactly as written (do not alter the functions):
(a) F(H) = {[A 0 B] • [(BC) Q D]}(H) with inputs as A(H), B(H), C(L), and
D(L). _
(b) K(L) = [A 0 C 0 (BD) 0 (ABCD)](L) with inputs from negative logic
sources.