Experiment#4 Boolean Algebra
1.
To verify the rules and regulations of Boolean Algebra.
2.
To simplify and modify Boolean logic functions by means of Demorgan’s theorem.
3.
To design and implement a logic circuit.
Boolean algebra is a deductive mathematical system closed over the values zero and one (false and true). A binary operator defined over this set of values accepts a pair of boolean inputs and produces a single boolean value.
P1: Boolean algebra is closed under the AND, OR, and NOT operations.
P2: The identity element with respect to • is one and + is zero. There is no identity
element with respect to logical NOT.
P3: The • and + operators are commutative.
P4: • and + are distributive with respect to one another. That is,
A • (B + C) = (A • B) + (A • C) and A + (B • C) = (A + B) • (A + C).
P5: For every value A there exists a value A’ such that A•A’ = 0 and A+A’ = 1.
This value is the logical complement (or NOT) of A.
P6:
• and + are both associative. That is, (A•B)•C = A•(B•C) and (A+B)+C = A+(B+C).
You can prove all other theorems in boolean algebra using these postulates.
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Experiment#4 Boolean Algebra
1. A+0 = A
2. A+1 = 1
3. A .0 = 0
4. A .1 = A
5. A+A = A
6. A+A’ = 1
7. A.A = A
8. A.A’ = 0
9. (A’)’ = A
10. A+AB = A
11. A+A’B = A+B
12. (A+B)(A+C) = A+BC
13. A’. B’ = (A+B)’
14. A’+B’ = (A.B)’
1.
Refer to your text book, understand the concepts of Boolean Algebra.
2.
Draw truth table, logic diagram and pin diagram for each part in THEORY .
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Experiment#4 Boolean Algebra
KL-31001 trainer kit, 7400 Quadruple 2 input NAND gates, 7402 Quadruple 2 input
NOR gates, 7408 Quadruple 2 input AND gates, 7432 Quadruple 2 input OR gates, 7404
Hex inverters.
Part I: a) Connect these circuits and verify their operations.
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Experiment#4 Boolean Algebra
Part II: Demorgan’s Theorem a) Proof of equation (1):
Construct the two circuits corresponding to the functions A’. B’and (A+B)’ respectively.
Show that for all combinations of A and B, the two circuits give identical results.
Connect these circuits and verify their operations. b) Proof of equation (2)
Construct two circuits corresponding to the functions A’+B’and (A.B)’ A.B, respectively.
Show that, for all combinations of A and B, the two circuits give identical results.
Connect these circuits and verify their operations.
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Experiment#4 Boolean Algebra
Part III: Design of a Digital Circuit
Consider the following problem:
Four chairs A, B, C, and D are placed in a row. Each chair may be occupied (“l”) or empty (“0”). A Boolean function F is “l” if and only if there are two or more adjacent chairs that are empty.
1. Give the truth table defining the Boolean function F
2. Express F as a minterm expansion (standard sum of product)
3. Express F as a maxterm expansion (standard product of sum)
4. Using theorems of Boolean algebra, simplify the minterm expansion of F to a form with as few occurrences of each as possible.
5. implement the simplified Boolean function with logic gates. Show the pin diagram.
Your task, is to design the circuitry of the logic system to open the waste valve if and only if there is good flame proven by the sensors.
The waste valve is opened if at least two out of the three sensors show good flame.
1. Give the truth table defining the Boolean function F
2. Express F as a minterm expansion (standard sum of product)
3. Express F as a maxterm expansion (standard product of sum)
4. Using theorems of Boolean algebra, simplify the minterm expansion of F to a form with as few occurrences of each as possible.
5. implement the simplified Boolean function with logic gates. Show the pin diagram.
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