Chapter 2- Boolean Algebra II PUC, MDRPUC, Hassan
Chapter-2
BOOLEAN ALGEBRA
Introduction:
An algebra that deals with binary number system is called “Boolean Algebra”.
It is very power in designing logic circuits used by the processor of computer system.
The logic gates are the building blocks of all the circuit in a computer.
Boolean algebra derives its name from the mathematician George Boole (1815-1864) who is
considered the “Father of symbolic logic”.
Boolean algebra deals with truth table TRUE and FALSE.
It is also called as “Switching Algebra”.
Binary Valued Quantities – Variable and Constants:
A variable used in Boolean algebra or Boolean equation can have only one of two variables.
The two values are FALSE (0) and TRUE (1)
A Sentence which can be determined to be TRUE or FALSE are called logical statements or
truth functions and the results TRUE or FALSE is called Truth values.
The variables which can store the truth values are called logical variables or binary valued
variables. These can store one of the two values 1 or 0.
The decision which results into either YES (TRUE or 1) or NO (FALSE or 0) is called Binary
decision.
Truth Table:
A truth table is a mathematical table used in logic to computer functional values of logical
expressions.
A truth table is a table whose columns are statements and whose rows are possible scenarios.
Example: Consider the logical expression
Logical Statement: Meals = “Ram prefer rice and roti for the meal”
Y = A AND B (Logical Variables: Y, A, B, Logical Operator AND)
Ram Prefer Rice Ram Prefer Roti Meals
FALSE FALSE FALSE
FALSE TRUE FALSE
TRUE FALSE FALSE
TRUE TRUE TRUE
1|Page Keerthi Kumar H.M
,Chapter 2- Boolean Algebra II PUC, MDRPUC, Hassan
If result of any logical statement or expression is always TRUE or 1, it is called Tautology and
if the result is always FALSE or 0, it is called Fallacy.
Logical Operators:
There are three logical operator, NOT, OR and AND.
These operators are now used in computer construction known as switching circuits.
NOT Operator:
The Not operator is a unary operator. This operator operates on single variable.
The operation performed by Not operator is called complementation.
The symbol we use for it is bar.
𝐗 means complementation of X
If X=1, X=0 If X=0, X=1
The Truth table and the Venn diagram for the NOT operator is:
X X
1 0
0 1
OR Operator:
The OR operator is a binary operator. This operator operates on two variables.
The operation performed by OR operator is called logical addition.
The symbol we use for it is ‘+’.
Example: X + Y can be read as X OR Y
The Truth table and the Venn diagram for the NOT operator is:
X Y X+Y
0 0 0
0 1 1
1 0 1
1 1 1
AND Operator:
The AND operator is a binary operator. This operator operates on two variables.
The operation performed by AND operator is called logical multiplication.
The symbol we use for it is ‘.’.
Example: X . Y can be read as X AND Y
2|Page Keerthi Kumar H.M
, Chapter 2- Boolean Algebra II PUC, MDRPUC, Hassan
The Truth table and the Venn diagram for the NOT operator is:
X Y X.Y
0 0 0
0 1 0
1 0 0
1 1 1
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.
o For two variables i.e. X, Y then truth table will need 22 or 4 rows.
o 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.
Example: Consider the following Boolean Expression F=X+ 𝐘
Step 1: This expression as two variables X and Y, then 22 or 4 rows.
Step 2: List the variables and every combination of X and Y.
Step 3: Create a new column 𝐘 of the statement, and then fill the truth values of Y in that
column.
Step 4: The final column contain the values of X+ 𝐘.
X Y Y X+ 𝐘
0 0 1 1
0 1 0 0
1 0 1 1
1 1 0 1
Exercise Problems:
1. Prepare a table of combination for the following Boolean algebra expressions.
a) X Y + XY b) XY Z + X Y Z
2. Verify using truth table for the following Boolean algebra.
a) X + XY = X b) X + Y = X . Y
3|Page Keerthi Kumar H.M
Chapter-2
BOOLEAN ALGEBRA
Introduction:
An algebra that deals with binary number system is called “Boolean Algebra”.
It is very power in designing logic circuits used by the processor of computer system.
The logic gates are the building blocks of all the circuit in a computer.
Boolean algebra derives its name from the mathematician George Boole (1815-1864) who is
considered the “Father of symbolic logic”.
Boolean algebra deals with truth table TRUE and FALSE.
It is also called as “Switching Algebra”.
Binary Valued Quantities – Variable and Constants:
A variable used in Boolean algebra or Boolean equation can have only one of two variables.
The two values are FALSE (0) and TRUE (1)
A Sentence which can be determined to be TRUE or FALSE are called logical statements or
truth functions and the results TRUE or FALSE is called Truth values.
The variables which can store the truth values are called logical variables or binary valued
variables. These can store one of the two values 1 or 0.
The decision which results into either YES (TRUE or 1) or NO (FALSE or 0) is called Binary
decision.
Truth Table:
A truth table is a mathematical table used in logic to computer functional values of logical
expressions.
A truth table is a table whose columns are statements and whose rows are possible scenarios.
Example: Consider the logical expression
Logical Statement: Meals = “Ram prefer rice and roti for the meal”
Y = A AND B (Logical Variables: Y, A, B, Logical Operator AND)
Ram Prefer Rice Ram Prefer Roti Meals
FALSE FALSE FALSE
FALSE TRUE FALSE
TRUE FALSE FALSE
TRUE TRUE TRUE
1|Page Keerthi Kumar H.M
,Chapter 2- Boolean Algebra II PUC, MDRPUC, Hassan
If result of any logical statement or expression is always TRUE or 1, it is called Tautology and
if the result is always FALSE or 0, it is called Fallacy.
Logical Operators:
There are three logical operator, NOT, OR and AND.
These operators are now used in computer construction known as switching circuits.
NOT Operator:
The Not operator is a unary operator. This operator operates on single variable.
The operation performed by Not operator is called complementation.
The symbol we use for it is bar.
𝐗 means complementation of X
If X=1, X=0 If X=0, X=1
The Truth table and the Venn diagram for the NOT operator is:
X X
1 0
0 1
OR Operator:
The OR operator is a binary operator. This operator operates on two variables.
The operation performed by OR operator is called logical addition.
The symbol we use for it is ‘+’.
Example: X + Y can be read as X OR Y
The Truth table and the Venn diagram for the NOT operator is:
X Y X+Y
0 0 0
0 1 1
1 0 1
1 1 1
AND Operator:
The AND operator is a binary operator. This operator operates on two variables.
The operation performed by AND operator is called logical multiplication.
The symbol we use for it is ‘.’.
Example: X . Y can be read as X AND Y
2|Page Keerthi Kumar H.M
, Chapter 2- Boolean Algebra II PUC, MDRPUC, Hassan
The Truth table and the Venn diagram for the NOT operator is:
X Y X.Y
0 0 0
0 1 0
1 0 0
1 1 1
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.
o For two variables i.e. X, Y then truth table will need 22 or 4 rows.
o 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.
Example: Consider the following Boolean Expression F=X+ 𝐘
Step 1: This expression as two variables X and Y, then 22 or 4 rows.
Step 2: List the variables and every combination of X and Y.
Step 3: Create a new column 𝐘 of the statement, and then fill the truth values of Y in that
column.
Step 4: The final column contain the values of X+ 𝐘.
X Y Y X+ 𝐘
0 0 1 1
0 1 0 0
1 0 1 1
1 1 0 1
Exercise Problems:
1. Prepare a table of combination for the following Boolean algebra expressions.
a) X Y + XY b) XY Z + X Y Z
2. Verify using truth table for the following Boolean algebra.
a) X + XY = X b) X + Y = X . Y
3|Page Keerthi Kumar H.M
