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Boolean Algebra

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A binary variable is one which can have two values only. For example: Is this day Tuesday? What can be the answers to this question. Only two. True or False, or maybe Yes or No. Boolean algebra is a study of mathematical operations of conjunction and disjunction on binary variables.

The two values for the binary variable can be taken as True or False, or 1 or 0. Conjunction and disjunction are denoted with the symbol $\vee$ and $\wedge$ respectively.
  A function in the form of f: $P^{k} \rightarrow P$, where the values of P = 0, 1 and value of ‘k’ is non- negative Integer is known as Boolean function. There are some basic rules and theorems which make the Boolean algebra work.

Definition

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Boolean algebra is used to find the algebra of real numbers but it is used for some numeric operations which is multiplication pq, addition p + q and negative values are replaced by some logical operations of conjunction, disjunction and negations. These are the operations of Boolean algebra.

Boolean algebra has different structures which are represented by 'Hasse diagram'. If a Boolean function has two binary operations i.e. logical AND or we can say “widge” and the notation of AND operation is "$\wedge$" and other operation is logical OR or we can say “vee”, the notation of ‘OR’ logical operation is “$'vee$” and both operations satisfy different type of laws such as idempotent law, absorption law, commutative law and distributive law. Both Boolean algebra and Boolean logic are same terms.

Symbols are used to represent operations in Boolean algebra. For the laws in Boolean algebra, any of these symbols can be used.

 Conjunction AND, $\cdot $, $\wedge$  
 Disjunction  OR, +, $\vee$
 Negation  NOT, '

Boolean algebra does follow these basic laws on the operators of algebra. They are given below:

1)  Idempotent laws: Suppose $P \wedge P$ = $P \vee P$ = P

2) Commutative laws: Let $P \wedge Q$ = $Q \wedge P$ ; And $P \vee Q$ = $Q \vee P$

3) Associative laws: $\rightarrow$ $P \wedge (Q \wedge R)$ = $(P \wedge Q) \wedge R$; $\rightarrow$ $P \vee (Q \vee R)$ = $(P \vee Q) \vee R$

4) Absorption laws: $\rightarrow$ $P \wedge (P \vee Q)$ = $P \vee (P \wedge Q) = P$

5) Distributive: $\rightarrow$ $P \wedge (Q \vee R)$ = $(P \wedge Q) \vee (P \wedge R)$ $\rightarrow$ $P \vee (Q \wedge R)$ = $(P \vee Q) \wedge (P \vee R)$
Universal laws are applicable on empty set $\phi $ and universal set I:

1) $\phi\wedge P=  \phi $

2) $\phi\vee  P=  P $

3) $I\wedge P=P $

4) $I\vee  P=  I$

Theorems

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Negation operator is also a major operator of Boolean algebra. The symbol for negation is '. These are some important laws for negation.

1) Involution law: (P')' = P

2) Complementary laws: The given laws are complementary laws.

i) P + P' = 1

ii) P$\cdot $P' = 0

The De-morgan theorem is one of the most important theorem of Boolean algebra.The De-Morgan's theorems are given here:

1) (P + Q + R + ...)' = P'Q'R'...

2) (PQR...)' = P' + Q' + R' + ...

Simplify

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Boolean Algebra can be considered as logical Calculus which is widely used in digital computer system.

 
Possibility of happening something may be either true of false. Boolean algebra is also based on true and false. True and false are pointed by ‘1’ and ‘‘0’ respectively in boolean algebra.

If there are two variables A and B then there are four possible combinations of occurrence:

A

B

False (0)

False (0)

False (0)

True (1)

True (1)

False(0)

True (1)

True (1)


The various laws and theorem expressed above can be used to simplify a Boolean expression. Process of simplifying boolean algebra problems is shown below with the help of an example:

Prove that: (P + Q)(PR + PR’) + PQ + Q = P + Q using simplification,.

Simplification:

Given (P + Q)(PR + PR’) + PQ + Q

= (P + Q) P (R + R’) + PQ + Q. (Distributive law).

= (P + Q) P + PQ + Q (Complementary law).

= P ((P + Q) + Q) + Q (Distributive law).

= P (P + Q) + Q (Idempotent law).

= PP + P Q + Q (Distributive law).

= P + (P + T) Q  (Universal law).

= P + Q.

Examples  

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Example 1: Simplify P + (PQ)'.

Solution: P + (PQ)' = P + P' + Q' (De-Morgan's law).

                = 1 + Q' (Complementary law) .

                = 1 (Universal law).

Example 2: Prove that (PQ)'(P' + Q) = P'.

Solution: (PQ)'(P' + Q)

               = (P' + Q')(P'+Q)

               = P'P' + P'Q + Q'P' + Q'Q (Distributive law).

               = P' + P'(Q + Q') (Distributive and complementary laws).

               = P' + P' (Complementary law).

               = P' (Idempotent law).