Page 149 - Engineering Digital Design

120 CHAPTER 3/BACKGROUND FOR DIGITAL DESIGN

[6] (A + B + AC) O(AB + C) = (A + B + AC)(AB + C) Corollary II
= (A + B + C)(AB + C) Absorption [Eqs. (3.13)]
= C + (A + B)(AB) Distributive law
[Eqs. (3.12)]
= C Factoring law; AND and
OR laws

[7] ac + (a + b} O (a + be) = ac + (ab) 0 (a + be) DeMorgan's law [Eqs. (3.15);
Eqs. (3.23)]
= ac + (ab) + (a + be) Corollary I [Eq. (3.25)]
= ab + a + be Factoring law; AND and OR laws
= a + b + be Absorption [Eqs. (3.13)]
= a + b + c Absorption

[8] wxy + wxz + wxz + wyz + xz = wxy + wxz + wxz Consensus law [Eqs. (3.14)]
= wxy + xz(w + w) Factoring law [Eqs. (3.12)]
= wxy + xz Or laws

[9] A 0 B 0 (A + B) = A 0 [B 0 (AB)] Eqs. (3.27)
= A 0 [B(l 0 A)] XOR Factoring law [Eqs. (3.19)]
= A 0 (AB) Eqs. (3.29)
= A(l 0 B) XOR Factoring law
= AB Eqs. (3.29)

[10] / = d 0 bed 0 abd 0 cd 0 ad 0 abed _
= [d® cd] 0 [abd 0 ad] 0 [bed 0 abed] Rearranging terms
= [d(l 0 c)] 0 [ad(b 0 1)] 0 [bcd(l 0 a)] XOR Factoring law [Eqs. (3.19)]
= cd 0 abd 0 abed Repeated applications of Eqs. (3.29)
= cd 0 [abd( 1 0 c)] XOR Factoring law [Eqs. (3.19)]
= cd 0 abed Application of Eqs. (3.29)
Notice that the gate/input tally of / has been reduced from 10/24 to 3/8 in the final
expression. Application of Corollary I further reduces / to (abc + cd).

FURTHER READING

Additional reading on the subject of mixed logic notation and symbology can be found in
the texts of Comer, Fletcher, Shaw and Tinder.

[1] D. J. Comer, Digital Logic and State Machine Design, 3rd ed. Saunders College Publishing, Fort
Worth, TX, 1995.