Boolean Algebra
Category : JEE Main & Advanced
(1) In Boolean algebra only two states of variables (0 and 1) are allowed.
(2) The variables (A, B, C ....) of Boolean Algebra are subjected to three operations.
(i) OR Operation : Represented by (+) sign
Boolean expression \[Y=A+B\]
When switch A or B is closed ? Bulb glows
(ii) AND Operation : Represented by \[(.)\] sign
Boolean expression \[Y=A\cdot B\]
When switches A and B both are closed ? Bulb glows
(iii) NOT Operation : Represented by bar over the variables
Boolean expression \[Y=\bar{A}\]
(3) Basic Boolean postulates and laws
(i) Boolean Postulates : \[0+A=A,\] \[1\,\,\cdot \text{ }A=A,\]
\[1+A=1,\]
\[0\cdot A=0,\]
\[A+\bar{A}=1\]
(ii) Identity law : \[A+A=A,\] \[A\cdot A=A\]
(iii) Negation law : \[\overline{{\bar{A}}}=A\]
(iv) Commutative law : \[A+B=B+A,\] \[A\cdot B=B\cdot A\]
(v) Associative law : \[(A+B)+C=A+(B+C),\]
\[(A\cdot B)\cdot C=A\cdot (B\cdot C)\]
(vi) Distributive law : \[A\cdot (B+C)=A\cdot B+A\cdot C\]
\[(A+B)\cdot (A+C)=A+BC\]
(vii) Absorption laws : \[A+A\cdot B=A,\] \[A\cdot (A+B)=A\]
\[\overline{A}\,\text{ }\!\!\cdot\!\!\text{ }\,(A+B\text{)}=\overline{A}\,\text{ }\!\!\cdot\!\!\text{ }\,B\]
(viii) Boolean identities : \[A+\overline{A}\,B=A+B\], \[A(\overline{A}+B)=AB\],
\[A+BC=(A+B)\,(A+C)\], \[(\overline{A}+B)\,\text{ }\!\!\cdot\!\!\text{ }\,\text{(}A+C)\,=\overline{A}C+AB\]
(ix) De Morgan's theorem : It states that the complement of the whole sum is equal to the product of individual complements and vice versa i.e.
\[\overline{A+B}=\bar{A}\cdot \bar{B}\] and \[\overline{A\cdot B}=\bar{A}+\bar{B}\]
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