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Boolean Algebra
67 questions By Tony R. Kuphaldt
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Question 31 of 67
Two very important rules of simplification in Boolean algebra are as follows:
- Rule 1: \(A + AB = A\)
- Rule 2: \(A+\overline{A}B = A+B\)
Not only are these two rules confusingly similar, but many students find them difficult to successfully apply to situations where a Boolean expression uses different variables (letters), such as here:
$$\overline{R}ST+\overline{R}$$
Here, it is the first rule that applies \((A + AB = A)\) and not the second rule \((A+A\overline{B}=A+B)\), giving a simplification of:
$$\overline{R}$$
Try to apply these two rules to the following Boolean expressions, identifying which rule directly applies, or if neither rule directly applies:
$$\overline{X} \ \overline{Y}Z+\overline{XY}$$
- $$FGH + G$$
$$\overline{C}+CF$$
$$\overline{AB}C+A$$
$$RS+\overline{R}$$
$$\overline{AB}+ABC$$
$$\overline{AB}C+C$$
$$\overline{R}V\overline{W}+\overline{R}$$
- $$\overline{J} \ \overline{K}LM+\overline{J}K$$
- $$\overline{E}HF+F\overline{E}$$
Reveal answer- $$FGH + G = G \ \ \ \ \ \ \ \ \ \ \ (Rule \ 1)$$
$$\overline{C}+CF=\overline{C}+F \ \ \ \ \ \ \ \ \ \ \ (Rule \ 2)$$
$$\overline{AB}C+A \ \ \ \ \ \ \ \ \ \ \ (Neither \ rule \ applies)$$
$$RS+\overline{R}=\overline{R}+S \ \ \ \ \ \ \ \ \ \ \ (Rule \ 2)$$
$$\overline{AB}+ABC=\overline{AB}+C \ \ \ \ \ \ \ \ \ \ \ (Rule \ 2)$$
$$\overline{AB}C+C=C \ \ \ \ \ \ \ \ \ \ \ (Rule \ 1)$$
$$\overline{R}V\overline{W}+\overline{R}= \overline{R} \ \ \ \ \ \ \ \ \ \ \ (Rule \ 1)$$
$$\overline{X} \ \overline{Y}Z+\overline{XY} \ \ \ \ \ \ \ \ \ \ \ (Neither \ rule \ applies)$$
- $$\overline{J} \ \overline{K}LM+\overline{J}K \ \ \ \ \ \ \ \ \ \ \ (Neither \ rule \ applies)$$
- $$\overline{E}HF+F\overline{E}= \overline{E}F \ \ \ \ \ \ \ \ \ \ \ (Rule \ 1)$$
Notes:Many students find the substitution of Boolean variables (going from the A’s and B’s of canonical rules to the different variables of real expressions where the rules are to be applied) very mysterious and difficult. Problems such as this give them practice learning to identify the rules’ patterns despite similarities or differences in the actual variables (letters) used.
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Question 32 of 67
Use Boolean algebra to simplify the following expression, then draw a logic gate circuit for the simplified expression:
$$A(B + AB) + AC$$
Reveal answer
Notes:Have your students explain the entire process they used in answering this question: simplifying the expression using Boolean algebra techniques, and developing a gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person’s technique (rather than just the instructor’s), which will allow them to see examples of how to do these processes cast in slightly different terms.
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Question 33 of 67
Use Boolean algebra to simplify the following expression, then draw a logic gate circuit for the simplified expression:
$$(A+B)(\overline{A}+\overline{B})$$
Reveal answer
Challenge question: identify the specific logic gate type that will perform this Boolean function using just a single gate.
Notes:Have your students explain the entire process they used in answering this question: simplifying the expression using Boolean algebra techniques, and developing a gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person’s technique (rather than just the instructor’s), which will allow them to see examples of how to do these processes cast in slightly different terms.
Question 29, answer 3 has a typo - should be A*B + !(D)*E not A*B+!(D*E)