Boolean Algebra and Logic Gates

• Boolean Algebra is named after George Boole, who developed it in 1854. • A binary logic variable is always 0 or 1, and in binary logic, 1+1=1, unlike in binary arithmetic where 1+1=10. • Logic gates are the basic building blocks of any digital system, and the inverter symbol (NOT gate) has a triangle that denotes an amplifier, and a circle at the output that denotes logical inversion. • DeMorgan's Theorem has two parts: the first theorem states that the complement of the sum of two or more variables is equal to the product of the complement of the variables, and the second theorem states that the complement of the product of two or more variables is equal to the sum of the complements of the variables. • DeMorgan's First theorem proves that an AND gate with inverted output is equivalent to an OR gate with inverted inputs, and can be represented as (A.B)' = A'+B'. • DeMorgan's Second theorem proves that an OR gate with inverted output is equivalent to an AND gate with inverted inputs, and can be represented as (A+B)' = A'.B'. • To obtain the DeMorgan equivalent for an AND, NAND, OR, or NOR gate, inverters (NOT-gates) are added to all inputs and outputs, and the AND symbol is changed to an OR symbol or vice versa. • Boolean reductions often result in an equation in one of two forms: Product of Sums (POS) or Sum of Product (SOP). • SOP expression is the most common and can be easily constructed using an AND-OR-INVERT gate (AOI). • A literal is a primed or unprimed variable, and if each term in SOP and POS forms contains all the literals, they are known as canonical SOP and POS, respectively. • Each individual term in canonical SOP and POS form is called a minterm and maxterm, respectively. • Minterms and maxterms can be represented using the X Y Z column, where for Minterm 0=x' and 1=x, and for Maxterm 0=x and 1=x'. • Any Boolean function can be written in Minterms and Maxterms, such as F(A,B,C) = Σ(1,4,5,6,7) = m1+m4+m5+m6+m7. • Digital circuits are broadly divided into two categories: Combinational logic, which has no memory and the present output depends only on the present input, and Sequential logic, which has memory and the present output depends on both the present and past inputs.