12 summary of basic boolean identities boolean

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Unformatted text preview: a effectively. Sr. No. Identities Dual identities 1 A+0=A A- 1 =A 2 A+l = 1 A-0 = 0 3 A«+A = A A-A = A 4 A + A =1 A- A =0 5 A=A 6 A+B=B+A 7 9 (A + B) + C = A + (B + (A • B) • C = A • (B • C) C) A(B + C) = AB + A-C A + B ■ C = (A + B) • (A + C) A+A-B=A A- (A + B) = A 10 A+ A -B = A + B A-(A+B) = A-B 11 A + B= A•B AB= A + B 8 AB=B-A Figure 6.12. Summary of basic Boolean identities. BOOLEAN FUNCTIONS A Boolean function is an expression formed with binary variables, the two binary operators OR and AND, the unary operator NOT, parentheses and equal sign. For a given value of the variables, the value of the function can be either 0 or 1. For example, consider the equation W = X+ Y Z Here the variable W is a function of X, Y and Z. This is written as W = f(X, Y, Z) and the right hand side of the equation is called an expression. The symbols X, Y and Z are referred to as literals of this function. The above is an example of a Boolean function represented as an algebraic expression. A Boolean function may also be represented in the form of a truth table. The number of rows in the table will be equal to 2 n, where n is the number of literals (binary variables) used in the function. The combinations of 0s and Is for each row of this table is easily obtained from the binary numbers by counting from 0 to 2" - 1. For each row of the table, there is a value for the function equal to either 0 or 1, which is listed in a separate column of the table. Such a truth table for the function W = X + Y • Z is shown in Figure 6.13. Observe that there are eight (23) possible distinct combinations for assigning bits to three variables. The column labeled W is either a 0 or a 1 for each of these combinations. The table shows that out of eight, there are five different combinations for which W = 1. X Y Z w u 0 0 0 1 1 1 1 0 0 1 1 -"0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 r l l l Figure 6.13. Truth table for the Boolean function W = X + Y • Z. The question now arises - is an algebraic expression for a given Boolean function unique? In other...
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