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190 CHAPTER 4 /LOGIC FUNCTION REPRESENTATION AND MINIMIZATION

order beginning with the ordinate (vertical) axis, as has been done throughout this text.
(a) P(A, B,C) = (A + B + C)(A + B + C)(A + B + C)(A + B + C)(A + B + C)
(b) 0(a,fc, c) = £>(!, 2, 4, 5, 6)
(c) W(a, b, c) = abc + abc + abc + abc + abc
(d) X(A, B,C) = Y\M(0,l ,2, 6,7)
(e) Y(w, x, y) = wx + xy + w(x © y) + wy (Hint: Expand first.)
(f) Z(A, 5, C) = (A + £) Q (A Q + AB (Hint: First construct a truth table
with input A.)
(g) F(X, Y, Z) = XY®YZ®XZ + XY [Hint: See Eq. (3.33).]
4.3 Place each of the four-variable functions below in a canonical truth table and in a
conventional ( 1 's and O's) K-map. Place the variables on the K-map axes in alphabetical
order beginning with the ordinate (vertical) axis, as has been done throughout this text.
(a) R(u, v, w, x) = £ m(0, 2, 3, 7, 8, 9, 10, 11, 13)
(b) S(a, b, c, d) = (a + b}(d + bc)(b + c)(a + b + c)
(c) T(W, X, Y, Z) = YZ + WXY + WXYZ + XYZ + WYZ + WXYZ + XYZ
(d) U(A,B,C,D) = Y\ M(0,5,8,9, 11, 12, 15)
(e) V(a, b, c,d) = J2 m(0, 4, 5, 7, 8, 9, 13, 15)
(f) W(u, v, w, x) = [(v + w) O x](u + w}(u + v)(u + x)
(g) X(A, B, C, D) = (A 0 B)CD + BCD + BCD + (A + B)CD + AB(C Q D)
(Hint: First construct a truth table for CD, then map the result into a 1's and O's
K-map.)
(h) F(W, x, Y, z) = (x e z) e [W(Y e z>] + XYZ
(Hint: First construct a truth table for WX, then map the result into a 1's and O's
K-map.)
4.4 Place each function of Problem 4. 1 into a conventional (1's and O's) K-map and extract
canonical (coded) SOP and POS expressions from that K-map.
4.5 Minimize each function of Problem 4.2 in both SOP and POS form with a third-order
K-map. By using the gate/input tally (exclusive of possible inverters) determine which
is simpler, the SOP or POS expression. Do not implement with logic gates.
4.6 Minimize each function of Problem 4.3 in both SOP and POS form with a fourth-
order K-map. By using the gate/input tally (exclusive of possible inverters), determine
which is simpler, the SOP or POS expression. Do not implement with logic gates.
4.7 The following three-variable functions are incompletely specified functions, that is,
they contain don't cares. By using a third-order K-map, minimize each function in
both SOP and POS form with and without the use of the don't cares in each case.
Identify any OPIs that may be present.
(a) <?(A,fl,C) = £>(0, 1,2, 7) + 0(3, 5)
(b) f(X , Y, Z) = 0 M(3, 4, 6) - 0(0, 2)
(c) g(fl,£>,c) = £>(0,l,5,7) + 0(2,4)
(d) h(x, y,z) = U M(3, 4, 5) - 0(0, 1, 2)
(e)