Page 172 - FUNDAMENTALS OF COMPUTER

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172 Fundamentals of Computers NPP

A A A B B B A + B A . B

0 0 0 0 0 0 1 1 1 1 1 1
0 0 0 1 1 1 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 0 0 0 0 0 0

We can see from the above truth table that CnamoŠV gË`Vm{bH$m go ñnï> h¡ {H$ V¥Vr` ñV§^ Ho$
the values in third column are exactly similar g^r _mZ MVwW© ñV§^ Ho$ g^r _mZm| Ho$ ~am~a h¡ Ÿ& AV…
to the values in forth column. Therefore we can
say that the expression in third column is equal XmoZm| ì`§OH$ ~am~a h¡ …
to the expression in forth column. Thus:

A B+ = A.B
Second Theorem {ÛVr` à_o`
Now, draw the truth table taking two input EH$ gË` Vm{bH$m ~ZmE {Og_| A VWm B d Xmo
variables A and B, and two Boolean functions ì`§OH$ A.B Am¡a A + hm|Ÿ&
B
A.B and A + B .
Proof: gË`mnZ:

A B B . A A + B
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 0

We can see that the expression in third CnamoŠV Vm{bH$m go ñnï> h¡ {H$ V¥Vr` ñV§^ Ho$
column is equal to the expression in forth g^r _mZ, MVwW© ñV§^ Ho$ g^r _mZm| Ho$ ~am~a h¡Ÿ&
column for all possible values of A and B. Thus,
A.B = A B+ . AV… A B . = A + B
The DeMorgan’s second theorem can be {S>_m°J©Z Ho$ {ÛVr` à_o` H$mo H$WZ Ho$ ê$n _|
stated in words as: Bg àH$ma go {bIm Om gH$Vm h¡…
“The complement of a logical product is “H$m°påßb_|Q> H$m `moJ, JwUZ\$b Ho$ H$m°påßb_|Q> Ho$
equal to the logical sum of complements.” ~am~a hmoVm h¡ Ÿ&”