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Theorems-Boolean algebra

Introduction

In Boolean Algebra, theorems are rules or properties that help simplify complex logic expressions easily.

By applying these theorems, we can:

  • Reduce the number of gates in circuits.
  • Design simpler and more efficient digital circuits.

List of Important Boolean Theorems

1. Idempotent Law

  • Statement:
    • A+A=AA + A = AA+A=A
    • A⋅A=AA \cdot A = AA⋅A=A
  • Meaning:
    • ORing or ANDing a value with itself gives the same value.
  • Example:
    • 0+0=00 + 0 = 00+0=0, 1+1=11 + 1 = 11+1=1
    • 0⋅0=00 \cdot 0 = 00⋅0=0, 1⋅1=11 \cdot 1 = 11⋅1=1

2. Identity Law

  • Statement:
    • A+0=AA + 0 = AA+0=A
    • A⋅1=AA \cdot 1 = AA⋅1=A
  • Meaning:
    • ORing with 0 or ANDing with 1 gives the original value.
  • Example:
    • 1+0=11 + 0 = 11+0=1, 0+0=00 + 0 = 00+0=0
    • 1⋅1=11 \cdot 1 = 11⋅1=1, 0⋅1=00 \cdot 1 = 00⋅1=0

3. Null (Dominance) Law

  • Statement:
    • A+1=1A + 1 = 1A+1=1
    • A⋅0=0A \cdot 0 = 0A⋅0=0
  • Meaning:
    • ORing with 1 always gives 1; ANDing with 0 always gives 0.
  • Example:
    • 0+1=10 + 1 = 10+1=1, 1+1=11 + 1 = 11+1=1
    • 0⋅0=00 \cdot 0 = 00⋅0=0, 1⋅0=01 \cdot 0 = 01⋅0=0

4. Complement Law

  • Statement:
    • A+A‾=1A + \overline{A} = 1A+A=1
    • A⋅A‾=0A \cdot \overline{A} = 0A⋅A=0
  • Meaning:
    • A value ORed with its complement is 1.
    • A value ANDed with its complement is 0.
  • Example:
    • If A=1A = 1A=1, then 1+0=11 + 0 = 11+0=1, 1⋅0=01 \cdot 0 = 01⋅0=0.

5. Involution Law

  • Statement:
    • A‾‾=A\overline{\overline{A}} = AA=A
  • Meaning:
    • Double complement of a value gives the original value back.
  • Example:
    • 1‾‾=1\overline{\overline{1}} = 11=1, 0‾‾=0\overline{\overline{0}} = 00=0

6. Commutative Law

  • Statement:
    • A+B=B+AA + B = B + AA+B=B+A
    • A⋅B=B⋅AA \cdot B = B \cdot AA⋅B=B⋅A
  • Meaning:
    • The order of variables does not matter in AND or OR operations.
  • Example:
    • 1+0=0+1=11 + 0 = 0 + 1 = 11+0=0+1=1
    • 1⋅0=0⋅1=01 \cdot 0 = 0 \cdot 1 = 01⋅0=0⋅1=0

7. Associative Law

  • Statement:
    • (A+B)+C=A+(B+C)(A + B) + C = A + (B + C)(A+B)+C=A+(B+C)
    • (A⋅B)⋅C=A⋅(B⋅C)(A \cdot B) \cdot C = A \cdot (B \cdot C)(A⋅B)⋅C=A⋅(B⋅C)
  • Meaning:
    • Grouping of variables does not affect the result.
  • Example:
    • (1+0)+1=1+(0+1)=1(1 + 0) + 1 = 1 + (0 + 1) = 1(1+0)+1=1+(0+1)=1

8. Distributive Law

  • Statement:
    • A⋅(B+C)=(A⋅B)+(A⋅C)A \cdot (B + C) = (A \cdot B) + (A \cdot C)A⋅(B+C)=(A⋅B)+(A⋅C)
    • A+(B⋅C)=(A+B)⋅(A+C)A + (B \cdot C) = (A + B) \cdot (A + C)A+(B⋅C)=(A+B)⋅(A+C)
  • Meaning:
    • Distribution of AND over OR and vice versa is allowed.
  • Example:
    • 1⋅(0+1)=(1⋅0)+(1⋅1)=0+1=11 \cdot (0 + 1) = (1 \cdot 0) + (1 \cdot 1) = 0 + 1 = 11⋅(0+1)=(1⋅0)+(1⋅1)=0+1=1

9. Absorption Law

  • Statement:
    • A+(A⋅B)=AA + (A \cdot B) = AA+(A⋅B)=A
    • A⋅(A+B)=AA \cdot (A + B) = AA⋅(A+B)=A
  • Meaning:
    • One variable can absorb terms involving itself.
  • Example:
    • 1+(1⋅0)=11 + (1 \cdot 0) = 11+(1⋅0)=1
    • 0⋅(0+1)=00 \cdot (0 + 1) = 00⋅(0+1)=0

10. De Morgan’s Theorems (VERY IMPORTANT)

There are two De Morgan’s Theorems:

  • First Theorem: A⋅B‾=A‾+B‾\overline{A \cdot B} = \overline{A} + \overline{B}A⋅B=A+B (The complement of AND is equal to OR of complements.)
  • Second Theorem: A+B‾=A‾⋅B‾\overline{A + B} = \overline{A} \cdot \overline{B}A+B​=A⋅B (The complement of OR is equal to AND of complements.)
  • Example:
    • 1⋅0‾=0‾=1\overline{1 \cdot 0} = \overline{0} = 11⋅0=0=1, and 1‾+0‾=0+1=1\overline{1} + \overline{0} = 0 + 1 = 11+0=0+1=1

Summary Table of Boolean Theorems

No.TheoremExpression
1IdempotentA+A=AA + A = AA+A=A, A⋅A=AA \cdot A = AA⋅A=A
2IdentityA+0=AA + 0 = AA+0=A, A⋅1=AA \cdot 1 = AA⋅1=A
3Null (Dominance)A+1=1A + 1 = 1A+1=1, A⋅0=0A \cdot 0 = 0A⋅0=0
4ComplementA+A‾=1A + \overline{A} = 1A+A=1, A⋅A‾=0A \cdot \overline{A} = 0A⋅A=0
5InvolutionA‾‾=A\overline{\overline{A}} = AA=A
6CommutativeA+B=B+AA + B = B + AA+B=B+A, A⋅B=B⋅AA \cdot B = B \cdot AA⋅B=B⋅A
7Associative(A+B)+C=A+(B+C)(A+B)+C = A+(B+C)(A+B)+C=A+(B+C), (A⋅B)⋅C=A⋅(B⋅C)(A\cdot B)\cdot C = A\cdot(B\cdot C)(A⋅B)⋅C=A⋅(B⋅C)
8DistributiveA⋅(B+C)=(A⋅B)+(A⋅C)A\cdot(B+C) = (A\cdot B)+(A\cdot C)A⋅(B+C)=(A⋅B)+(A⋅C)
9AbsorptionA+(A⋅B)=AA+(A\cdot B)=AA+(A⋅B)=A, A⋅(A+B)=AA\cdot(A+B)=AA⋅(A+B)=A
10De Morgan’sA⋅B‾=A‾+B‾\overline{A \cdot B} = \overline{A}+\overline{B}A⋅B=A+B, A+B‾=A‾⋅B‾\overline{A+B} = \overline{A}\cdot\overline{B}A+B​=A⋅B

Conclusion

  • Boolean theorems make it possible to simplify complicated logic circuits into simpler ones.
  • They are used to minimize the number of gates, thus saving cost, space, and power.
  • Mastering these theorems is essential for anyone working in computer science, electronics, or IT fields.