Boolean Algebra Summary

• 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 xB, there exists x' such that x + x' = 1 and x . x' = 0.

If B = {0, 1} in the above definition, B is a Boolean algebra.

• A simple proposition (statement) is a sentence which in a given content can said to be either true or false.

Truth value of proposition is taken as either true or false.

• Compound proposition is a combination of two or more simple propositions connected by the connectivity 'AND', 'OR', 'if then' and 'if and only if'.

• Conditional proposition pq is false only when p is true and q is false. In other cases, it is true.

If pq, then the contrapositive of this proposition is -q -p.

• The biconditional pq is true only when both p and q are true or both p and q are false.

• A compound proposition which is always true for all combination of truth values of its components is called a tautology.

• A compound proposition which is always false for all combination of truth values of its components is called a contradiction.

• 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.

Example:

x.y' + x.z + y.z'

A Boolean expression can be regarded as a Boolean function since it has a unique value, either 0 or 1.

• 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.

• The logical gates can be expressed in Boolean expressions.

• The Boolean expression help in making complex switching circuits.

• A complicated combinatorial circuits which has a Boolean expression can be simplified by simplifying the Boolean expression.