Page 171 - FUNDAMENTALS OF COMPUTER

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NPP Number System, Boolean Algebra and Logic Circuits 171

12. A.(B.C) = (A.B).C
Proof: The values of these expressions are Proof: gyÌ Z§. 11 Ho$ AZwgma XmoZm| ì`§OH$m| Ho$
found to be equal for all possible values of A, {bE gyÌ H$s Vm{bH$m ~ZmH$a CZHo$ _mZ kmV {H$E Om
B and C. This can be proved by drawing a table
similar to that of identity 11. gH$Vo h¡Ÿ&
13. A.(B + C) = A.B + A.C
Proof: A Table can be drawn similar to the Proof: gyÌ 11 Ho$ g_mZ Vm{bH$m ~ZmH$a XmoZm|
table drawn in identity No. 11. We can find that ì`§OH$m| H$mo ~am~a {gÕ {H$`m Om gH$Vm h¡… AV… A, B d
the two expressions have same values for all C Ho$ g^r g‘yhm| Ho$ {bE XmoZm| 춧OH$m| Ho$ ‘mZ g‘mZ h¢&
possible combinations of A, B and C.
14. A + B.C = (A + B) . (A + C)
Proof: Take R.H.S.
(A + B) . (A + C) = A.A + A.C + B.A + B.C
= A + A.C + A.B + B.C
= A(1 + C + B) + B.C
= A.1 + B.C
= A + B.C = L.H.S.
Thus, R.H.S. = L.H.S. Hence Proved.

15. A = A
Proof:

Put A = , 0 A = 0 = = 0 = A
1
put A = , 1 A = = 0 = = A
1
1
3.13 De Morgan’s Theorems 3.13 {S> _m°J©Z Ho$ à_o`
These theorems are used in algebraic BZH$m Cn`moJ {H$gr ~y{b`Z \$bZ Ho$ ~rOJ{UVr`
simplification of Boolean functions. There are gabrH$aU _| hmoVm h¡Ÿ& {S>_m°J©Z Ho$ {ZåZ Xmo à_o` h¢…
two Demorgan’s theorems:
1. A B+ = A.B

2. A.B = A + B
First Theorem àW_ à_o`

Consider first theorem A B+ = A.B . The Bg ~y{b`Z g_rH$aU H$mo H$WZ Ho$ ê$n _| Bg
Boolean equation can be stated as “The àH$ma {bIm Om gH$Vm h¡… “{H$gr `moJ H$m H$m°påßb_|Q>,
complement of a logical sum is equal to the ~am~a hmoVm h¡, H$m°påßb_|Q> Ho$ JwUZ\$b Ho$ Ÿ&”
product of complements.”
Proof of Theorem I: Draw the truth table àW_ à_o` H$m gË`mnZ: Xmo BZnwQ> d¡[aE~b A VWm
taking two input variables A and B and getting B boH$a Ed§ Xmo \§$ŠeZ A + d A.B Ûmam gË`
B
the two functions A + B and A.B .
Vm{bH$m ~Zm`|Ÿ&