This document discusses Boolean algebra and logic gates. It defines Boolean algebra as having two values - True or False. Logic circuits in computers are also designed to have two states - high (1) or low (0). The three basic Boolean operators are AND, OR, and NOT. Truth tables represent all possible combinations of variable values in a Boolean expression. Common logic gates like AND, OR, and NOT are used to implement Boolean functions and are the building blocks of digital circuits. DeMorgan's theorems relate Boolean operations and their complements.

1. Lecture no : 12
Introduction to Computer
Mr. Muhammad Moazzam
Faculty of DHPT,
University of the Lahore.

2. Boolean Algebra

3. Boolean Algebra
• In Boolean Algebra , elements have one of
two values –True or False or (0,1)
• The circuits in a computer are also designed
for two-state operations
• That is input and output of a circuit is either
low(0) or high(1)
• The circuits are called logic circuits.

4. BOOLEAN VARIABLES
• The variables in Boolean Algebra can
take only two values- True or false.
• The variables are called Boolean
variables.

5. Boolean Operators
• There are three basic operators in Boolean
Algebra which are called logical operators or
Boolean operators.
OR - logical addition
AND – logical multiplication
NOT – Logical negation

6. TRUTH TABLES
• A truth table is a table that represents the
possible values of the operands and
corresponding values of a Boolean operation
or a Boolean expressions.
• Boolean expression with ‘n’ number of
variables , the truth table will have 2n rows.

7. LOGIC GATES
• A logic gate is an electronic circuit which
makes logic decisions.
• A logic gate takes one or more inputs and
will produce only one output.
• Logic gates are the building blocks from
which most of the digital systems are built
up.

8. • The three basic logical operators are
OR,AND and NOT
• They are said to be logically complete as any
Boolean function can be realized in terms of
these connectives .
• Gate used to implement these logical operators
are known as basic logic gates.
LOGIC GATES

9. EVALUATION OF BOOLEAN EXPRESSION USING
TRUTH TABLE
To create a truth table, follow the steps given below.
 Step 1: Determine the number of variables, for n
variables create a table with 2n rows.
 For two variables i.e. X, Y then truth table will need 22 or
4 rows.
 For three variables i.e. X, Y, Z, then truth table will need
23 or 8 rows.
 Step 2: List the variables and every combination of 1
(TRUE) and 0 (FALSE) for the given variables
 Step 3: Create a new column for each term of the
statement or argument.
 Step 4: If two statements have the same truth values,
then they are equivalent. 13

10. OR Operation
• Boolean expression for the OR operation:
x =A + B
• The above expression is read as
“x equals A OR B”

11. OR Gate
• An OR gate is a gate that has two or more
inputs and whose output is equal to the OR
combination of the inputs.

12. AND Gate
• The and gate is a logic circuit which
accepts two or more input signals but
produces only one output signal.
• The output signal produced will be 1 only if
all input signals are 1 otherwise it will be 0.

13. AND Operation
• Boolean expression for the AND operation:
x =A.B
• The above expression is read as
“x equals AAND B”

14. AND Gate
• An AND gate is a gate that has two or more
inputs and whose output is equal to the
AND product of the inputs.

15. NOT Gate
• The NOT gate is a logic circuit which will
accept only one input signal .
• The output state produced by NOT gate will be
always the opposite of the input signal
• Hence it is called the inverter.

16. NOT Operation
• The NOT operation is an unary operation, taking only
one input variable.
• Boolean expression for the NOT operation:
x = A
• The above expression is read as “x equals the inverse
of A
• Also known as inversion or complementation.
• Can also be expressed as: A’

17. NOT Operation

18. NAND Gate
 Not AND Gate - NOT + AND gate It is a combination
of NOT and AND gates It is Complemented AND
Gate
 Symbol of NAND gate

19. NAND Gate

20. NAND Gate

21. NOR Gate

22. NOR Gate

23. Boolean algebra
• A Boolean algebra comprises...
– A set of elements
– Binary operators {+ , •} Boolean sum and
product
– A unary operation { ' } (or { }) example: A’ or A

24. Boolean algebra
• …and the following axioms
– The set B contains at least two elements {a b} with a  b
– Closure: a+b is in B a•b is in B
– Commutative: a+b = b+a a•b = b•a
– Associative: a+(b+c) = (a+b)+c a•(b•c) = (a•b)•c
– Identity: a+0 = a a•1 = a
– Distributive: a+(b•c)=(a+b)•(a+c) ,a•(b+c)=(a•b)+(a•c)
– Complementarity: a+a' = 1 a•a' = 0

25. DeMorgan's theorem
 First Theorem:
 The De-Morgan's first theorem states that, "The
complement of a sum equals to the product of the
complements".
 i.e. (A+B)'=A'.B’

26. DeMorgan's
theorem

27. DeMorgan's theorem
Second Theorem:
De Morgan's second theorem states that, "The
complement of a product is equal to the sum of the
complements."
i.e. (A.B)'=A'+B'

28. DeMorgan's theorem