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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. Moreover, for all x B, there exists x' such that x + x' = 1 and x . x' = 0. |
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| If B = {0, 1} in the above definition, B is a Boolean algebra. |
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A simple proposition (statement) is a sentence which in a given content can said to be either true or false. |
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| Truth value of proposition is taken as either true or false. |
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Compound proposition is a combination of two or more simple propositions connected by the connectivity 'AND', 'OR', 'if then' and 'if and only if'. |
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Conditional proposition p q is false only when p is true and q is false. In other cases, it is true. |
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If p q, then the contrapositive of this proposition is -q  -p. |
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The biconditional p q is true only when both p and q are true or both p and q are false. |
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A compound proposition which is always true for all combination of truth values of its components is called a tautology. |
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A compound proposition which is always false for all combination of truth values of its components is called a contradiction. |
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A Boolean expression consists of variables x, y, z …etc where these variables take the values 0 or 1. They may be connected by ' . ', '+' or ' operation. |
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| Example: |
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| x.y' + x.z + y.z' |
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| A Boolean expression can be regarded as a Boolean function since it has a unique value, either 0 or 1. |
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The postulates, theorems or Boolean expressions in Boolean algebra hold good when the operation '+' and '.', the symbols '1' and '0' are interchanged, this is called principle of duality. If a theorem is proved, then a dual theorem is as good as proved. |
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The logical gates can be expressed in Boolean expressions. |
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The Boolean expression help in making complex switching circuits. |
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A complicated combinatorial circuits which has a Boolean expression can be simplified by simplifying the Boolean expression. |
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