Boolean Algebra and Logic Gates: Fundamentals of Digital Logic, Schemes and Mind Maps of Electronics

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Chapter-

Boolean Algebra & Logic Gates

Boolean Algebra & Logic Gates:

Boolean variables, Basic laws, De Morgan’s Theorems, Basic

Gates, Universal gates, XOR and XNOR gates

Boolean algebra

• The signal in most present day electronic digital system uses just two discrete values and are therefore said to be binary.
• Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854).
• George boole introduced the concept of binary number system in the studies of the mathematical theory of logic and developed its algebra known as Boolean algebra.

Boolean Algebra Laws

Idempotent Law a. a = a a + a = a

Associative law a. (b. c) = (a. b). c a + (b + c) = (a + b) + c

Commutative law a. b = b. a a + b = b + a

Distributive law a. (b + c) = a. b + a. c a + (b. c) = (a + b). (a + c)

De-Morgan law (a + b)’ = a’. b’ (a. b)’ = a’ + b’

Identity law a + 0 = a a. 0 = 0 a + 1 = 1 a. 1 = a

Complementation law 0’ = 1 1’ = 0 a. a’= 0 a+ a’ = 1

Involution law (a’)’ = a

Q. The idempotent law in Boolean algebra says that:

(NET-JUNE-2008)

(A) ∼(∼x)=x

(B) x+x=x

(C) x+xy=x

(D) x(x+y)=x

Q: A ∨ A = A is called: (NET-DEC-2004)

(A) Identity law

(B) De Morgan’s law

(C) Idempotent law

(D) Complement law

Electromagnetic Relays

Pneumatic Logic

Not Gate(Inverter)

• It represents not logical operator is also known as inverter, it is a unary operator, which simply complement the input.

Truth Table Input Output

X Y = X’ 0

1

• OR gate satisfy all the 3 rules idempotent, associative, and commutative. 1. Idempotent Law: a + a = a 1. Associative law: a + (b + c) = (a + b) + c 1. Commutative law: a + b = b + a

And Gate

• It is a digital logic gate, that implements logical conjunction. Output will be high if and only if all input are high otherwise low.

Truth Table Input Output A B Y = A. B 0 0 0 1 1 0 1 1

Q. The output expression of the following gate network is

(ISRO-2011)

(A) X. Y + X’ Y’

(B) X. Y + X. Y

(C) X. Y

(D) X + Y

Nor gate

• The output will be high if and only if all inputs are low. Or simply a OR gate followed by an inverter.
• NOR gate is also called universal gate because it can be used to implement any other logic gate. We will cover this property extensively in the next chapter. Truth Table Input Output A B Y = (A + B)’ 0 0 0 1 1 0 1 1

• Idempotent and associative law

• (a + a)’ a

• ((a + b)’+ c)’ (a + (b + c)’)’

• Commutative law.

• (a + b)’ (b + a)’

NAND Gate

• The output will be low if and only if all inputs are high. Or simply an and gate followed by an inverter
• NAND gate is also called universal gate because it can be used to implement any other logic gate. We will cover this property extensively in the next chapter.

Truth Table Input Output A B Y = (A. B)’ 0 0 0 1 1 0 1 1