3. Logic Circuits
A collection of individual logic gates connect with
each other and produce a logic design known as a
Logic Circuit
The following are the types of logic circuits:
Decision making
Memory
A gate has two or more binary inputs and single output.
Fundamentals of Digital Electronics - Logic Gates 3
4. Basic Logic Gates
The following are the three basic gates:
NOT
AND
OR
Each logic gate performs a different logic function. You
can derive logical function or any Boolean or logic
expression by combining these three gates.
Fundamentals of Digital Electronics - Logic Gates 4
5. NOT Gate
The simplest form of a digital logic circuit is the
inverter or the NOT gate
It consists of one input and one output and the
input can only be binary numbers namely; 0 and 1
the truth table for NOT Gate:
Fundamentals of Digital Electronics - Logic Gates 5
6. AND Gate
The AND gate is a logic circuit that has two or more inputs
and a single output
The operation of the gate is such that the output of the gate
is a binary 1 if and only if all inputs are binary 1
Similarly, if any one or more inputs are binary 0, the output
will be binary 0.
Fundamentals of Digital Electronics - Logic Gates 6
7. OR Gate
The OR gate is another basic logic gate
Like the AND gate, it can have two or more inputs and a
single output
The operation of OR gate is such that the output is a binary
1 if any one or all inputs are binary 1 and the output is
binary 0 only when all the inputs are binary 0
Fundamentals of Digital Electronics - Logic Gates 7
8. NAND Gate
The term NAND is a contraction of the expression
NOT-AND gate
A NAND gate, is an AND gate followed by an
inverter
The algebraic output expression of the NAND gate
is Y = A.B
Fundamentals of Digital Electronics - Logic Gates 8
9. NOR Gate
The term NOR is a contradiction of the expression
NOT-OR
A NOR gate, is an OR gate followed by an inverter
The algebraic output expression of the NOR gate is
Y = A + B
Fundamentals of Digital Electronics - Logic Gates 9
10. EX-OR and EX-NOR Gates
EX-OR and EX-NOR are digital logic circuits that may
use two or more inputs
EX-NOR gate returns the output opposite to EX-OR
gate
EX-OR and EX-NOR gates are also denoted by XOR
and XNOR respectively.
Fundamentals of Digital Electronics - Logic Gates 10
11. EXOR Gate
The Ex-OR (Exclusive- OR) gate returns high output with one of
two high inputs (but not with both high inputs or both low
inputs)
For example, if both the inputs are binary 0 or 1, it will return the
output as 0. Similarly, if one input is binary 1 and another is
binary 0, the output will be 1 (high)
The operation for the Ex-OR gate is denoted by encircled plus
symbol
The Ex-OR operation is widely used in digital circuits.
The algebraic output expression of the Ex-OR gate is Y = A ⊕ B =
Fundamentals of Digital Electronics - Logic Gates 11
12. EXNOR Gate
The Ex-NOR (Exclusive- NOR) gate is a circuit that returns low
output with one of two high inputs (but not with both high inputs)
For example, if both the inputs are binary 0 or 1, it will return the
output as 1. Similarly, if one input is binary 1 and another is binary 0,
the output will be 0 (low)
The symbol for the Ex-NOR gate is denoted by encircled plus symbol
which inverts the binary values
The algebraic output expression of the Ex-NOR gate is Y =
Fundamentals of Digital Electronics - Logic Gates 12
13. Applications of Logic Gates
The following are some of the applications of Logic
gates:
Build complex systems that can be used to different fields
such as
Genetic engineering,
Nanotechnology,
Industrial Fermentation,
Metabolic engineering and
Medicine
Construct multiplexers, adders and multipliers.
Perform several parallel logical operations
Used for a simple house alarm or fire alarm or in the circuit
of automated machine manufacturing industry
Fundamentals of Digital Electronics - Logic Gates 13
14. Boolean Laws
Three of the basic laws of Boolean
algebra are the same as in ordinary
algebra; the commutative law, the
associative law and the distributive law
The commutative law for addition and
multiplication of two variables is
written as,
A + B = B + A
A . B = B . A
Fundamentals of Digital Electronics - Logic Gates 14
15. Associative law
This law states that the order in which the
logic operations are performed is
irrelevant as their effect is the same.
The associative law for addition and
multiplication of three variables is
written as,
(A + B) + C = A + (B + C)
(A .B) . C = A. (B. C)
Fundamentals of Digital Electronics - Logic Gates 15
16. Distributive law
Distributive law states the following
condition.
The distributive law for three variables
involves, both addition and multiplication
and is written as,
A (B+ C) = A B + AC
Note that while either '+' and „.‟ s can be
used freely.
The two cannot be mixed without
ambiguity in the absence of further rules.
Fundamentals of Digital Electronics - Logic Gates 16
17. For example does
A . B + C means (A . B) + C or A . (B+ C)?
These two form different values for A = O,
B = 1 and C = 1, because we have
(A . B) + C = (0.1) + 1 = 1
A . (B + C) = 0 . (1 + 1) = 0
which are different. The rule which is
used is that „.‟ is always performed before
'+'. Thus X . Y + Z is (X.Y) + Z.
Fundamentals of Digital Electronics - Logic Gates 17
18. Basic Duality in Boolean Algebra:
We state the duality theorem without proof. Starting
with a Boolean relation, we can derive another
Boolean relation by 1.
Changing each OR (+) sign to an AND (.) sign 2.
Changing each AND (.) sign to an OR (+) sign. 3.
Complementary each 0 and 1 For instance
A +0= A The dual relation is A . 1 = A
Also since A (B + C) = AB + AC by distributive law.
Its dual relation is A + B C = (A + B) (A + C)
Fundamentals of Digital Electronics - Logic Gates 18
20. Theorems of Boolean Algebra
9) A = A
10) A + AB = A
11) A + AB = A + B
12) (A + B)(A + C) = A + BC
13) Commutative : A + B = B + A
AB = BA
14) Associative : A+(B+C) =(A+B) + C
A(BC) = (AB)C
15) Distributive : A(B+C) = AB +AC
(A+B)(C+D)=AC + AD + BC + BD
21. De Morgan’s Theorems
16) (X+Y) = X . Y
17) (X.Y) = X + Y
• Two most important theorems of Boolean
Algebra were contributed by De Morgan.
• Extremely useful in simplifying expression in
which product or sum of variables is inverted.
• The TWO theorems are :
22. Implications of De Morgan’s Theorem
(a) Equivalent circuit implied by theorem (16)
(b) Alternative symbol for the NOR function
(c) Truth table that illustrates DeMorgan’s Theorem
(a)
(b)
Input Output
X Y X+Y XY
0 0 1 1
0 1 0 0
1 0 0 0
1 1 0 0
(c)
23. Implications of De Morgan’s Theorem
(a) Equivalent circuit implied by theorem (17)
(b) Alternative symbol for the NAND function
(c) Truth table that illustrates DeMorgan’s Theorem
(a)
(b)
Input Output
X Y XY X+Y
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 0
(c)
24. De Morgan’s Theorem Conversion
Step 1: Change all ORs to ANDs and all ANDs to Ors
Step 2: Complement each individual variable
(short overbar)
Step 3: Complement the entire function (long overbars)
Step 4: Eliminate all groups of double overbars
Example : A . B A .B. C
= A + B = A + B + C
= A + B = A + B + C
= A + B = A + B + C
= A + B
25. De Morgan’s Theorem Conversion
ABC + ABC (A + B +C)D
= (A+B+C).(A+B+C) = (A.B.C)+D
= (A+B+C).(A+B+C) = (A.B.C)+D
= (A+B+C).(A+B+C) = (A.B.C)+D
= (A+B+C).(A+B+C) = (A.B.C)+D
