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NPP Number System, Boolean Algebra and Logic Circuits 169

3.12 Basic Identities in Boolean Algebra 3.12 ~y{b`Z ~rOJ{UV _| _yb^yV Cnà_o`
Identities are the relationships between Cnà_o` {H$Ýht Ma am{e`m| d ì`§OH$m| Ho$ ~rM
single variables or expressions which are used Eogo à_o` hmoVo h¢ {OZH$s ghm`Vm go {H$gr ~‹S>o ~y{b`Z
to minimise a Boolean expression. We will see
the identities and the proof for each identity: ì`§OH$ H$mo gab {H$`m Om gH$Vm h¡Ÿ& BZHo$ ñdê$n
VWm {dñV¥V {ddaU Bg àH$ma h¢…
1. A + 0 = A
Proof. When, A = 0, A + 0 = 0
A = 1, A + 0 = 1
Thus, for all values of A, the result is same AV … A Ho$ àË`oH$ _mZ Ho$ {bE n[aUm_ A Ho$ ~am~a
as A. hr hmoVm h¡ Ÿ&
2. A.0 = 0
Proof. When A = 0, A.0 = 0
When, A = 1, A.0 = 0
The result is zero for all values of A. A Ho$ g^r _mZm| Ho$ {bE n[aUm_ 0 h¡ Ÿ&
3. A + 1 = 1 NPP
Proof. When, A = 1, A + 1 = 1
A = 0, A + 1 = 1
Thus, the result is one for all values of A. AV … A Ho$ g^r _mZm| Ho$ {bE n[aUm_ 1 h¡ Ÿ&
4. A.1 = A
Proof. When, A = 0, A.1 = 0
A = 1, A.1 = 1
The result is same as the value of A. A Ho$ g^r _mZm| Ho$ {b`o n[aUm_ A h¡ Ÿ&
5. A + A = A
Proof. When, A = 0, A + A = 0
A = 1, A + A = 1
Thus the result is always same as A. A Ho$ g^r _mZm| Ho$ {bE n[aUm_ A h¡ Ÿ&
6. A.A = A
Proof. When A = 0, A.A = 0
A = 1, A.A = 1
The result is always same as A. AV… n[aUm_ ha ~ma A Ho$ ~am~a h¡ Ÿ&
7. A A1+ =
Proof. When A = 0, A A1+ =
A = 1, A A1+ =
The result is always one for all values of A. AV… n[aUm_ ha ~ma 1 Ho$ ~am~a h¡Ÿ&

8. A.A 0=
Proof. When A = 0, A.A 0=

A = 1, A.A 0=
Thus the result is zero for all values of A. AV… n[aUm_ ha ~ma 0 h¡Ÿ&