Facebook
Google
GitHub
LinkedinDigital Circuits
Boolean Algebra
67 questions By Tony R. Kuphaldt
-
Question 22 of 67
Complete the truth tables for these two Boolean expressions:
$$Output = \overline{A}+B$$
A B Output
0 0
0 1
1 0
1 1
$$Output =A+\overline{A}B$$
A B Output
0 0
0 1
1 0
1 1
Reveal answer$$Output =\overline{A}+B$$
A B Output
0 0 1
0 1 1
1 0 0
1 1 1
$$Output =A+\overline{A}B$$
A B Output
0 0 0
0 1 1
1 0 1
1 1 1
Notes:Ask your students to explain exactly how they figured out the “Output” states to fill in the blanks in the truth tables, for the different input combinations. Ask them also to compare and contrast this process with that of figuring out the truth table for a given logic gate circuit.
-
Question 23 of 67
Complete the truth tables for these two Boolean expressions:
$$Output =\overline{A}+\overline{B}+C$$
A B C Output
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
$$Output =A(B+AC+\overline{A})$$
A B C Output
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Reveal answer$$Output =\overline{A}+\overline{B}+C$$
A B C Output
0 0 0 1
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
$$Output =A(B+AC+\overline{A})$$
A B C Output
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
Notes:Ask your students to explain exactly how they figured out the “Output” states to fill in the blanks in the truth tables, for the different input combinations. Ask them also to compare and contrast this process with that of figuring out the truth table for a given logic gate circuit.
It is especially educational if you ask your students to suggest techniques for quickly determining truth table states, based on certain features of the Boolean expression. For instance, there is a way we can tell the first four “Output” states in the truth table (reading top to bottom) will be 0 without having to plug values into the expression for B and C. Discuss with your students how we can look at the expression, seeing A as a multiplier for the sum within the parentheses, and immediately conclude that half of the truth table outputs will be 0.
-
Question 24 of 67
Like real-number algebra, Boolean algebra is subject to the laws of commutation, association, and distribution. These laws allow us to build different logic circuits that perform the same logic function.
For each of the equivalent circuit pairs shown, write the corresponding Boolean law next to it:
Note: the three short, parallel lines represent ëquivalent to” in mathematics.
Reveal answerIn order, from top to bottom:
$$AB = BA$$
$$(AB)C = A(BC)$$
$$(A+B)C = AC+BC$$
$$A+B = B+A$$
$$(A+C)B = AB+CB$$
$$(A+B)+C = A+(B+C)$$
Notes:The commutative, associative, and distributive laws of Boolean algebra are identical to the respective laws in real number algebra. These should not be difficult concepts for your students to understand. The real benefit of working through these examples is to associate gate and relay logic circuits with Boolean expressions, and to see that Boolean algebra is nothing more than a symbolic means of representing electrical discrete-state (on/off) circuits. In relating otherwise abstract mathematical concepts to something tangible, students build a much better comprehension of the concepts.
Question 29, answer 3 has a typo - should be A*B + !(D)*E not A*B+!(D*E)