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Each premise has the subject-predicate form, and each shares a common element called the middle term. Switch to new thesaurus. Aristotelian logic - the syllogistic logic of Aristotle as developed by Boethius in the Middle Ages. Exact, valid, and rational reasoning: ratiocination , rationality , reason. What is sound or reasonable: rationale , rationality , rationalness , reason , sense. It is logical to assume that you will get a higher salary if you are promoted; She is always logical in her thinking. Mentioned in?

References in periodicals archive? Saha and Manna both JIS College of Engineering, India introduce engineering students to the classical methods of the design of digital circuits, providing background to and incorporating modern logic design techniques. Digital principles and logic design.

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We know that the development of 21 st century skills is crucial for workplace readiness and economic growth but how does this translate into real options for ambitious students and school leavers? Sign up to receive updates from McGraw-Hill:. Our Digital Success Consultant team is committed to helping instructors deploy a personalized learning and teaching experience—resulting in a successful course experience for instructors and, ultimately, greater student engagement and performance. The map has been. The two-variable Karnaugh map is a useful way to represent any of the 16 Boolean functions of two variables as described in section 3.

As an example, the function AB has been shown in Figure 4. Since the function AB is equal to the minterm m3, a 1 is placed in the cell corresponding to m3.

Figure 4. The minterm m3, for example, is assigned in the square corresponding to row 0 and column 11, thus making the binary number Note that, each of the variables has four squares where its logic value is 0 and four squares with logic value 1. To understand the usefulness of the map for simplifying the Boolean functions, we must observe the basic properties of the adjacent squares. Any two adjacent squares in the Karnaugh map differ by only one variable, which is complemented in one square and uncomplemented in one of the adjacent squares.

For example, in Figure 4. Example 4. First, a three-variable Karnaugh map is drawn and 1s are placed at the squares according to the minterms of the function as shown in Figure 4. A 1 1 Figure 4. The Karnaugh map for this function is shown in Figure 4. There are four squares marked with 1s, each for one of the minterms of the function.

Note that these 1s or minterms can be combined to produce a reduced term. Here the B variable is changing its form, from uncomplemented to complemented. As in the previous examples, it is shown that two adjacent squares consisting of 1s can be combined to form reduced terms. Let us consider the next example.

The Karnaugh map is shown in Figure 4. Here, it may observed that two of the variables A and B are changing their forms form uncomplemented to complemented. Therefore, these variables can be removed to form the reduced expression to C. Note that squares that are already considered in one group, can be combined with other group or groups. Here, the minterms are given by their decimal-equivalent numbers.

The same is redrawn in Figure 4.

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The minterm of a particular square can be obtained by combining the row and column. Different four-variable Karnaugh maps can be redrawn, if the variables are assigned an other way. Two, four, or eight adjacent squares can be combined to reduce the number of literals in a function. The squares of the top and bottom rows as well as leftmost and rightmost columns may be combined. For example, m0 and m2 can be combined, as can m4 and m6, m12 and m14, m8 and m10, m0 and m8, m1 and m9, m3 and m11, and m2 and m Similarly, the four squares of the corners i.

When two adjacent squares are combined, it is called a pair and represents a term with three literals. Four adjacent squares, when combined, are called a quad and its number of literals is two.


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If eight adjacent squares are combined, it is called an octet and represents a term with one literal. If, in the case all sixteen squares can be combined, the function will be reduced to 1. The Karnaugh map for the above expression is shown in Figure 4. Note that the reduced expression is not a unique one, because if pairs are formed in different ways as shown in Figure 4.

But both expressions are logically correct.

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The canonical SOP form for the above expression can be obtained as follows. The Karnaugh map for the above function is shown in Figure 4. One octet and two quads are formed. To obtain the minterms for the above expression, it needs to be expanded to the canonical SOP form as below.

One pair and two quads are formed one quad consists of the four squares of the corners. After forming the Karnaugh map, SOP expression can be realized as above. Four pairs are formed. It may be noted that one quad can also be formed, but it is redundant as the squares contained by the quad are already covered by the pairs which are essential. The above expression is given in respect to the maxterms. In the Karnaugh map, 0s are to placed instead of 1s at the corresponding maxterm squares. There are two ways to achieve the minimized expression above.

One way to is consider the 0s of the Karnaugh map. One octet and one quad has been formed with 0s. Two quads are formed considering the 1s. Two quads and a pair are formed considering the 1s of the Karnaugh map. Three quads are formed considering the 0s of the Karnaugh map.


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  5. The number of cells or squares becomes excessively large and combining the adjacent squares becomes complex. The number of cells or squares is always equal to the number of minterms. Figures 4.

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    AB m24 m25 m27 m26 m30 m31 m29 m In addition to those, while making groups or combinations, in Figures 4. Similarly, according to Figures 4. CD m3 m7 m15 m11 m27 m31 m23 m Similar to the method described above, six-variable Karnaugh maps are formed with 64 minterms as demonstrated in Figure 4. ABC m56 m57 m59 m58 m62 m63 m61 m DEF m7 m15 m31 m23 m55 m63 m47 m This is done to form groups like pairs, quadoctet, etc.

    In the Karnaugh map of Figure 4. These terms are called prime implicants. Form the set of prime implicants, essential prime implicants are determined by preparing a prime implicants chart. The minterms that are not covered by the essential prime implicants, are taken into consideration by selecting some more prime implications to obtain an optimized Boolean expression. Each minterm of the function is expressed by its binary representation. Each set of minterms possessing the same index are separated by lines.

    Now each of the minterms is compared with the minterms of a higher index. For each pair of terms that can combine, the new terms are formed. A line is drawn in when all the minterms of one set is compared with all the minterms of a higher index. The same process is repeated for all the groups of minterms. The process is to be continued until no new match is possible. All the terms that remain unchecked i. After obtaining the prime implicants, a chart or table is prepared where rows are represented by the prime implicants and the columns are represented by the minterms of the function.

    Crosses are placed in each row to show the composition of the minterms that makes the prime implicants. A completed prime implicant table is to be inspected for the columns containing only a single cross. Prime implicants that cover the minterms with a single cross are called the essential prime implicants. The table in Figure 4. Column I consists of the decimal equivalent of the function or the minterms and column II is the corresponding binary representation.