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# Boolean Algebra

Top
 Sub Topics A function in the form of f: Pk ⇢ P, where the values of P = 0, 1 and value of ‘k’ is non- negative Integer is known as Boolean function. Now we will see what is Boolean Algebra? Both Boolean algebra and Boolean logic are same terms. It 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 and 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 “∧” and other operation is logical OR or we can say “vee”, the notation of ‘OR’ logical operation is “∨” and both operations satisfy different type of laws which are shown below: Idempotent laws: Suppose P ∧ P = P ∨ P = P, These logical operations also satisfy the commutative laws: Let P ∧ Q = Q ∧ P; And P ∨ Q = Q ∨ P; Now we will see how it satisfies the associative laws: ⇒ P ∧ (Q ∧ R) = (P ∧ Q) ∧ R; ⇒ P ∨ (Q ∨ R) = (P ∨ Q) ∨ R; The operation satisfies the absorption laws: ⇒ P ∧ (P ∨ Q) = P ∨ (P ∧ Q) = P; The operations are manually distributive: ⇒ P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R); ⇒ P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R); ‘P’ contains universal bounds ∅ (empty Set) and I (the universal set) which satisfy: ⇒ ∅ ∧ P = ∅; ⇒ ∅ ∨ P = P; ⇒ I ∧ P = P; ⇒ I ∨ P = I; This is all about Boolean algebra.

## Simplify Boolean Algebra

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)

Following rules are used to simplify boolean algebra.
1) Write X + 0 = X which indicates that addition of zero to any number gives actual number. In place of zero, if we add one then it gives one (X + 1 = 1). When we use a complimentary number, X', and add, it also gives one (X + X' = 1) and when we add same Numbers then X + X = X.

2) Boolean identities for multiplication are used to solve any of the multiplication given in the expression. (0) (X) = 0, (1) (X) = X, (X) (X) = X, and (X) (X') = 0.

3) Also look (X + Y) as same to (Y + X). We can write (X) (Y) as (Y) (X). One more formula is used to simplify the given expression as X + (Y + Z) = (X + Y) + Z.

4) Some more formulas can also be used in simplification of Boolean expression.

X + XY = X which can be obtained by using above explained identities.
Process of simplifying boolean algebra poblems is shown below:
(P + Q)(PR + PR’) + PQ + Q = P + Q,
Simplification:
Given (P + Q)(PR + PR’) + PQ + Q,
= (P + Q) P (R + R’) + PQ + Q,
= (P + Q) P + PQ + Q,
= P ((P + Q) + Q) + Q,
= P (P + Q) + Q,
= PP + P Q + Q,
= P + (P + T) Q,
= P + Q.