The laws of Boolean expression can be defined axiomatically as certain equations called axioms together with their logical consequences called theorems, or semantically as those equations that are true for every possible assignment of 0 or 1 to their variables. The axiomatic approach is sound and complete in the sense that it proves respectively neither more nor fewer laws than the semantic approach.The Boolean algebra is branch of mathematics in which all the terms takes only the values 0 and 1.Here 0 represents true value and 1 represents false value. Moreover, perform the operations of addition logical AND, and multiplication logical OR. These operations are based on logical combinations. In Boolean algebra, alphabetic and arithmetic symbols are used as logical variable.

Definition of boolean expressions

A Boolean expression is defined as the expression in which constituents have one of two rates and the algebraic processes defined on the set are logical OR, a type of addition, and logical AND, a type of multiplication.

Boolean expression varies from the ordinary expression in three ways: the values that variables may presume, which are a logical instead of a numeric quality, perfectly 0 and 1; in the processes related to these values; and in the properties of those processes, that is, the laws that they follow. Expressions contain mathematical logical, digital logics, computer programming, set theory, and statistics.

Laws of Boolean algebra:

Laws of Boolean algebra:

(1a) x · y = y · x

(1b) x + y = y + x

(1c) 1 + x = 1

(2a) x · (y · z) = (x · y) · z

(2b) x + (y + z) = (x + y) + z

(3a) x · (y + z) = (x · y) + (x · z)

(3b) x + (y · z) = (x + y) · (x + z)

(4a) x · x = x

(4b) x + x = x

(5a) x · (x + y) = x

(5b) x + (x · y) = x

(6a) x · x1 = 0

(6b) x + x1 = 1

(7) (x1)1 = x

(8a) (x · y)1 = x1 + y1

(8b) (x + y) 1 = x1 · y1

Thus the above rules are used to translate the logic gates directly. Although some of the rules can be derived from simpler identities derived in the packages.

Example for Boolean Expressions:

Example 1: Determine the OR operations using Boolean expressions.

Solution: Construct a truth table for OR Operation,

x y x V y

F F F

F T T

T F T

T T T

Example 2: Determine the AND operations using Boolean expressions.

Solution: Construct a truth table for AND Operation,

x y x Λ y

F F F

F T F

T F F

T T T

Example3 : Determine the NOT operations using Boolean expressions.

Solution: Construct a truth table for NOT Operation,

x ¬x

F T

T F

Learn to solve examples for boolean expression:

We use the above learnt laws of boolean algebra to solve these problems:

1.) Prove that the Boolean expression rule x · (x + y) = x

Solution: x · (x + y) = x · x + x · y using 3.(a)

= x + x · y using 4.(a)

= x · (1 + y) using 3.(a)

= x · 1 using Exercise 1

= x as required.

Hence, the given Boolean expression is proved

2.) Prove that the Boolean expression: x + (x. y) = x

Solution: x + (x . y) = x (1+y) using 3.(a)

= x.1 using Exercise 1

= x

Hence, the given Boolean expression is proved

3.) Prove that the Boolean expression:

Solution: x + (y. z) = (x + y). (x +z)

= x .x + x .z + y .x + y .z ; we know, x .x =x

= x + x .z + x .y + y .z ; x + x .z = x using 5(a)

= x + x .y + y .z ; x+ x .y = x

= x + y .z

Hence, the given Boolean expression is proved.

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