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Boolean Algebra
67 questions By Tony R. Kuphaldt
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Question 52 of 67
Write the Boolean expression for this TTL logic gate circuit, then reduce that expression to its simplest form using any applicable Boolean laws and theorems. Finally, draw a new gate circuit diagram based on the simplified Boolean expression that performs the exact same logic function.
Reveal answerOriginal Boolean expression: \(\overline{AB+AC}\)
Reduced gate circuit:
Notes:The Boolean simplification for this particular problem is tricky. Remind students that complementation bars act as grouping symbols, and that parentheses should be used when in doubt to maintain grouping after “breaking bars” with DeMorgan’s Theorem.
Ask your students to compare the “simplified” circuit with the original circuit. Are any advantages apparent to the version given in the answer? Certainly, the Boolean expression for that version of the circuit is simpler compared to that of the original circuit, but is the circuit itself significantly improved?
This question underscores an important lesson about Boolean algebra and logic simplification in general: just because a mathematical expression is simpler does not necessarily mean that the expression’s physical realization will be any simpler than the original!
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Question 53 of 67
Suppose you needed an inverter gate in a logic circuit, but none were available. You do, however, have a spare (unused) NAND gate in one of the integrated circuits. Show how you would connect a NAND gate to function as an inverter.
Use Boolean algebra to show that your solution is valid.
Reveal answer
For the above solution: \(\overline{AA} = \overline{A}\)
Follow-up question: are there any other ways to use a NAND gate as an inverter? The method shown above is not the only valid solution!
Notes:Not only is the method shown in the answer not the only valid solution, but it may even be the worst one! Your students should be able to research or invent alternative inverter connections, so after asking them to present their alternatives, ask the class as a whole to decide which solution is better. Ask them to consider electrical parameters, such as propagation delay time and fan-out.
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Question 54 of 67
Suppose you needed an inverter gate in a logic circuit, but none were available. You do, however, have a spare (unused) NOR gate in one of the integrated circuits. Show how you would connect a NOR gate to function as an inverter.
Use Boolean algebra to show that your solution is valid.
Reveal answer
For the above solution: \(\overline{A+A} = \overline{A}\)
Follow-up question: are there any other ways to use a NOR gate as an inverter? The method shown above is not the only valid solution!
Notes:Not only is the method shown in the answer not the only valid solution, but it may even be the worst one! Your students should be able to research or invent alternative inverter connections, so after asking them to present their alternatives, ask the class as a whole to decide which solution is better. Ask them to consider electrical parameters, such as propagation delay time and fan-out.
Question 29, answer 3 has a typo - should be A*B + !(D)*E not A*B+!(D*E)