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Electronic Labs_12

Electronic Labs_12 - X-OR operation and the condition...

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Unformatted text preview: X-OR operation, and the condition whereby a true result is obtained when both variables are true is the distinguishing feature between the X-NOR and NOR operations. The X-NOR operation having two variables can be implemented using the logic circuit shown in Fig. A-7 (a). Other configurations are possible depending on the logic components available, but the logic symbol of (b) is commonly used to indicate the X-NOR function regardless of the type of Circuit. It can be seen from the logic circuit how the X-NOR function is produced. For example, if variables A and B both are true or false simulta- neously, the inverters insure that the inputs to both AND gates have opposite states. Therefore, both AND gate outputs are 0, which is inverted to 1 by the output NOR gate. If input variables A and B have opposite states, the inverters insure that one or the other of the AND gates has 1’s on both inputs. This results in a 1 output from the affected AND gate, which is inverted to 0 by the output NOR gate. The truth table of (c) shows all possible combinations of the X-NOR gate input-output conditions. The expanded version of the two variable X-NOR equation is stated as follows: X=A§+EB Mathematical manipulation of 3111.5. equation will produce the result X = A-B + A-B, which can be verified by the truth table of Fig. A-7 (c). This expression is sometimes referred to as an equality statement and its circuit is referred to as an equality comparator. The condensed X-NOR equation form is: X=A€BB mu:- X-NOR A .— B (:53 Fig. A-7 (c) Laws of Boolean Algebra Boolean logic statements rarely consist of single Appendix A A-5 “ operations. They more commonly involve a number of operations having two or more variables. This makes them difficult to visualize and the deductive analysis process employed for basic operations becomes cumbersome, if not impossible, as the statements increase in complexity. In order to implement Boolean statements mathematically, whether simple or complex, laws have been developed using basic expressions, the validity of which can be proved by logical analysis. These laws are termed the commutative, associative and distributive laws and have identical counterparts in ordinary mathematics. The logical AND (-) and OR (+) connectives take on the connotation of mathematical product and sum operators in these laws, but this is done so that the principles of algebra be applied to logic statements. Commutative Law. This law states that the result of a product (AND) or sum (OR) operation is unaffected by the sequence of the elements that make up the expression. For a two element‘expres- sion using the variables A and B, two important postulates can be stated: Postulate 13 A+B = B+A Postulate 14 AB = B-A These postulates indicate that as long as the same logical connective is used between a given group of elements, the positions of the elements on either side of the equal sign may be interchanged without affecting the result. Associative Law. This law states that the elements of a logic expression may be grouped in any quantity provided they are connected by the same sign. Two postulates evolve from this for the OR and AND operations: Postulate15 A+(B+C)=(A+B)+C Postulate 16 A-(B-C) = (A-B)-C In other words, it makes no difference how the elements are grouped as long as the AND and OR connectives are not changed. Note: In logic notation involving multiple elements of the AND operation it is common practice to omit the operational sign (-) and write the terms together (e.g. ABC). Where convenience dictates, this practice should be followed in order to simplify the statement of AND expressions. Distributive Law. This law states that if the sum of two or more quantities (multiplicand) is multiplied by another quantity (multiplier), the product is equal to the sum of the products obtained by multiplying each term of the multiplicand by the multiplier. This law is purely mathematical in ...
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