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Boolean Algebra
67 questions By Tony R. Kuphaldt
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Question 61 of 67
Shown here is the ladder logic diagram for a fire alarm system, where the activation of any alarm switch opens that (normally-closed) switch contact and sounds the alarm:
Write the Boolean expression for this relay circuit, then simplify that expression using DeMorgan’s Theorem and draw a new relay circuit implementing the simplified expression.
Reveal answerOriginal circuit expression:
$$\overline{\overline{A} \ \overline{B} \ \overline{C} \ \overline{D} \ \overline{E}}$$
Simplified expression and circuit:
$$A+B+C+D+E$$
Follow-up question: which circuit (the original or the one show above) is more practical from a fail-safe standpoint? In other words, which circuit will give the safest result in the event of a switch or wiring failure?
Notes:Here students see that even though two circuits are functionally identical (at least according to their respective Boolean expressions), they may not behave quite the same under adverse conditions (i.e. faulted switches or wiring). This is a very important thing for them to see, because it underscores the practical need to look beyond the immediate design criteria (Boolean function) and consider other parameters (failure mode).
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Question 62 of 67
The Law of Distribution in boolean algebra is identical to the law of distribution in “normal” algebra:
$$A(B + C) = AB + AC \ \ \ \ \ \ \ \ \ \ \ \ \ Applying \ the \ Law \ of \ Distribution$$
While the process of distribution is not difficult to understand, the reverse of distribution (called factoring) seems to be a more difficult process for many students to master:
$$AB + AC = A(B + C) \ \ \ \ \ \ \ \ \ \ \ \ \ Factoring \ A \ out \ of \ each \ term$$
Survey the following examples of factoring, and then describe what this process entails. What pattern(s) are you looking for when trying to factor a Boolean expression?
$$CD + AD + BD = D(C + A + B)$$
$$X\overline{Y} \ \overline{Z}+\overline{X} \ \overline{Y} Z = \overline{Y}(X\overline{Z}+\overline{X}Z)$$
$$J + JK = J(1 + K)$$
$$AB + ABCD + BCD + B = B(A + ACD + CD + 1)$$
Reveal answerWhen factoring, you must look for variables common to each product term.
Follow-up question: if implemented with digital logic gates, which of these two expressions would require the fewest components?
$$A(B + C)$$
$$AB + AC$$
Notes:Factoring really does seem to be a more difficult pattern-recognition skill to master than distribution, the latter being self-explanatory to many students. The purpose of this question is to get students to recognize and articulate the pattern-matching process involved with factoring. Once students have a working explanation of how to factor (especially if phrased in their own words), they will be better equipped to do so when needed.
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Question 63 of 67
Simplify this logic gate circuit, which uses nothing but NAND gates to accomplish a certain logic function:
Reveal answer
Notes:This question stands as an example of how NAND gates may be used to construct different types of logic functions. In fact, with a sufficient quantity of NAND gates, any logic function may be built. This is why NAND gates are said to be “universal.”
Question 29, answer 3 has a typo - should be A*B + !(D)*E not A*B+!(D*E)