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Introduction |
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The development of Boolean algebra is a more general theory of set theory and mathematical logic.
An English mathematician, named George Boole invented this new kind of algebra which analyses logic mathematically. This Boolean algebra provided a basic logic for operation on binary numbers 0, 1. Since computers are based on binary system, this branch of mathematics is found to be exactly useful for the internal working of various computers.
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Boolean Algebra as an Algebraic Structure |
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Boolean Algebra is an algebraic structure defined by a set of elements B, together with two operations, + and . satisfying the following axioms (Hunington postulates). |
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Principle of Duality |
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The dual of any statement in a Boolean algebra is the statement obtained by interchanging + and ., and simultaneously inter-changing the elements 0 and 1 in the statement. |
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Principle of Duality (Contd...) |
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An expression consisting of combinations of binary operations +, . and unary operation and a finite number of elements of a Boolean algebra is called a Boolean function. |
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Conditional and Biconditional Statements |
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If p and q are two propositions, then the compound proposition, "if p then q" is known as conditional statement or implication. |
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Conditional and Biconditional Statements (Contd...) |
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If two simple statements p and q are connected by the connective 'if and only if', then the resulting compound statement is called the biconditional statement. |
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Arguments and their Validity |
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A compound proposition is a tautology if it is always true for all possible combination of the truth values of its components.
If a compound proposition which is always false for all possible combination of the truth values, of its components then it is called a contradiction.
The argument (P1, P2, P3.....Pn;Q) is said to be valid if Q is true whenever all P1, P2, P3.....Pn are true. |
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Application to Switching Circuits |
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The application of Boolean algebra to electronic devices such as computers, lies in the restriction of the variable to two possible condition 'On and Off' or 'True or False' or numerically '1 or 0'. The electric circuits carry out the Boolean logic. |
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Application to Switching Circuits (Contd...) |
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Other Application Switching Circuits includes Logic Gates, AND Gate, OR Gate, NOT Gate, Combinatorial Circuit, Equivalent Circuit, |
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Summary |
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A Boolean algebra is a set B with two distinct elements, along with the binary operations '+' and '.' and a unary operation (') which satisfy closure property, commutative property, existence of unit element, distributive property for both the binary operations. |
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Conclusion |
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In this chapter, we have learnt the axioms of Boolean algebra. Also, we have learnt about Boolean expressions, their properties and their applications to switching circuits, in particular.
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