MODULE 3
Boolean Algebra, Fundamentals of Truth tables and Precedence
Algebra
Algebra means reunion on broken parts. It is the study of mathematical symbols and rules for
manipulating the symbols. Algebra can be regarded as elementary, abstract or modern depending
on the level or field of study.
Algebra has computations similar to arithmetic but with letters standing for numbers which
allows proofs of properties that are true regardless of the numbers involved. For example,
quadratic equation: ax2 + bx + c = 0 where a, b, c can be any number (a≠0). Algebra is used in
many studies, for example, elementary algebra, linear algebra, Boolean algebra, and so on.
1.1 Polynomials
A polynomial involves operations of addition, subtraction, multiplication, and non-negative
integer exponents of terms consisting of variables and coefficients. For example, x2 + 2x − 3 is a
polynomial in the single variable x. Polynomial can be rewritten using commutative, associative
and distributive laws.
An important part of algebra is the factorization of polynomials by expressing a given
polynomial as a product of other polynomials that cannot be factored any further. Another
important part of algebra is computation of polynomial greatest common divisors. x2 + 2x − 3 can
be factored as (x − 1)(x + 3).
1.2 Boolean Algebra
Boolean algebra is the branch of algebra in which the values of the variables are true values
denoted by 1 and 0 or true and false respectively.
Boolean algebra can be used to describe logic circuit; it is also use to reduce complexity of
digital circuits by simplifying the logic circuits. Boolean algebra is also referred to as Boolean
logic. It was developed by George Boole sometime on the 1840s and is greatly used in
computations and in computer operations. The name Boolean comes from the name of the
author.
Boolean algebra is a logical calculus of truth values. It somewhat resembles the arithmetic
algebra of real numbers but with a difference in its operators and operations. Boolean operations
involve the set {0, 1}, that is, the numbers 0 and 1. Zero [0] represents “false” or “off” and One
[1] represents “true” or “on”.
1 – True, on
, 0 – False, off
This has proved useful in programming computer devices, in the selection of actions based on
conditions set.
Basic Boolean operations
1. AND
The AND operator is represented by a period or dot in-between the two operands e.g
- X .Y
The Boolean multiplication operator is known as the AND function in the logic domain;
the function evaluates to 1 only if both the independent variables have the value 1.
2. OR
The OR operator is represented by an addition sign. Here the operation + is different from
that defined in normal arithmetic algebra of numbers. E.g. X+Y
The + operator is known as the OR function in the logic domain; the function has a value
of 1 if either or both of the independent variables has the value of 1.
3. NOT
The NOT operator is represented by X' or X̅.
This operator negates whatever value is contained in or assigned to X. It changes its value
to the opposite value. For instance, if the value contained in X is 1, X' gives 0 as the
result and if the value stored in X is 0, X' gives 1 as the result. In some texts, NOT may
be represented as X̅
To better understand these operations, truth table is presented for the result of any of the
operations on any two variables.
Truth Tables
A truth table is a mathematical table used in logic to compute the functional values of
logical expressions on each of their functional arguments. It is specifically in connection with
Boolean algebra and Boolean functions. Truth tables can be used to tell if a proposition
expression is logically valid. In a truth table, the output is completely dependent on the input. It
is composed of a column for each input entry and another column the corresponding output.
Each row of the truth table therefore contains one possible configuration of the input variables
(for instance, X=true Y=false), and the result of the operation for those values.
Applications of truth table
1. The truth table can be used in analyzing arguments.
2. It is used to reduce basic Boolean operations in computing
Boolean Algebra, Fundamentals of Truth tables and Precedence
Algebra
Algebra means reunion on broken parts. It is the study of mathematical symbols and rules for
manipulating the symbols. Algebra can be regarded as elementary, abstract or modern depending
on the level or field of study.
Algebra has computations similar to arithmetic but with letters standing for numbers which
allows proofs of properties that are true regardless of the numbers involved. For example,
quadratic equation: ax2 + bx + c = 0 where a, b, c can be any number (a≠0). Algebra is used in
many studies, for example, elementary algebra, linear algebra, Boolean algebra, and so on.
1.1 Polynomials
A polynomial involves operations of addition, subtraction, multiplication, and non-negative
integer exponents of terms consisting of variables and coefficients. For example, x2 + 2x − 3 is a
polynomial in the single variable x. Polynomial can be rewritten using commutative, associative
and distributive laws.
An important part of algebra is the factorization of polynomials by expressing a given
polynomial as a product of other polynomials that cannot be factored any further. Another
important part of algebra is computation of polynomial greatest common divisors. x2 + 2x − 3 can
be factored as (x − 1)(x + 3).
1.2 Boolean Algebra
Boolean algebra is the branch of algebra in which the values of the variables are true values
denoted by 1 and 0 or true and false respectively.
Boolean algebra can be used to describe logic circuit; it is also use to reduce complexity of
digital circuits by simplifying the logic circuits. Boolean algebra is also referred to as Boolean
logic. It was developed by George Boole sometime on the 1840s and is greatly used in
computations and in computer operations. The name Boolean comes from the name of the
author.
Boolean algebra is a logical calculus of truth values. It somewhat resembles the arithmetic
algebra of real numbers but with a difference in its operators and operations. Boolean operations
involve the set {0, 1}, that is, the numbers 0 and 1. Zero [0] represents “false” or “off” and One
[1] represents “true” or “on”.
1 – True, on
, 0 – False, off
This has proved useful in programming computer devices, in the selection of actions based on
conditions set.
Basic Boolean operations
1. AND
The AND operator is represented by a period or dot in-between the two operands e.g
- X .Y
The Boolean multiplication operator is known as the AND function in the logic domain;
the function evaluates to 1 only if both the independent variables have the value 1.
2. OR
The OR operator is represented by an addition sign. Here the operation + is different from
that defined in normal arithmetic algebra of numbers. E.g. X+Y
The + operator is known as the OR function in the logic domain; the function has a value
of 1 if either or both of the independent variables has the value of 1.
3. NOT
The NOT operator is represented by X' or X̅.
This operator negates whatever value is contained in or assigned to X. It changes its value
to the opposite value. For instance, if the value contained in X is 1, X' gives 0 as the
result and if the value stored in X is 0, X' gives 1 as the result. In some texts, NOT may
be represented as X̅
To better understand these operations, truth table is presented for the result of any of the
operations on any two variables.
Truth Tables
A truth table is a mathematical table used in logic to compute the functional values of
logical expressions on each of their functional arguments. It is specifically in connection with
Boolean algebra and Boolean functions. Truth tables can be used to tell if a proposition
expression is logically valid. In a truth table, the output is completely dependent on the input. It
is composed of a column for each input entry and another column the corresponding output.
Each row of the truth table therefore contains one possible configuration of the input variables
(for instance, X=true Y=false), and the result of the operation for those values.
Applications of truth table
1. The truth table can be used in analyzing arguments.
2. It is used to reduce basic Boolean operations in computing
