Math Problem Statement

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Define Boolean algebra and establish the following results: (i) a.(a+b) = a (ii) (a.b).c = a.(b.c) = (a.c).(b.c)

Solution

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Definition of Boolean Algebra:

Boolean algebra is a branch of mathematics dealing with variables that have two possible values: true (1) and false (0). It is commonly used in computer science, logic, and digital circuits. Boolean algebra uses three primary operations:

  1. AND (⋅): A ⋅ B is true only if both A and B are true.
  2. OR (+): A + B is true if either A or B is true.
  3. NOT ('): Negation inverts the value (true becomes false, and false becomes true).

Boolean algebra satisfies specific laws and properties:

  • Commutative law: A + B = B + A, A ⋅ B = B ⋅ A
  • Associative law: (A + B) + C = A + (B + C), (A ⋅ B) ⋅ C = A ⋅ (B ⋅ C)
  • Distributive law: A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C)
  • Identity laws: A + 0 = A, A ⋅ 1 = A
  • Idempotent laws: A + A = A, A ⋅ A = A
  • Complement laws: A + A' = 1, A ⋅ A' = 0

Solutions:

(i) Prove: a(a+b)=aa \cdot (a + b) = a

Using the laws of Boolean algebra:

  1. Apply the Distributive Law: a(a+b)=(aa)+(ab)a \cdot (a + b) = (a \cdot a) + (a \cdot b).
  2. From the Idempotent Law, aa=aa \cdot a = a, so a(a+b)=a+(ab)a \cdot (a + b) = a + (a \cdot b).
  3. From the Absorption Law, a+(ab)=aa + (a \cdot b) = a.

Thus, a(a+b)=aa \cdot (a + b) = a.


(ii) Prove: (ab)c=a(bc)=(ac)(bc)(a \cdot b) \cdot c = a \cdot (b \cdot c) = (a \cdot c) \cdot (b \cdot c)

We need to show the equality of these terms using associativity and commutativity:

  1. Start with (ab)c(a \cdot b) \cdot c:

    • Using the Associative Law, (ab)c=a(bc)(a \cdot b) \cdot c = a \cdot (b \cdot c).
  2. For a(bc)=(ac)(bc)a \cdot (b \cdot c) = (a \cdot c) \cdot (b \cdot c):

    • By the Distributive Law, we can rewrite a(bc)a \cdot (b \cdot c) as combining terms aca \cdot c and bcb \cdot c, since these terms are commutative.

Thus, (ab)c=a(bc)=(ac)(bc)(a \cdot b) \cdot c = a \cdot (b \cdot c) = (a \cdot c) \cdot (b \cdot c).


Let me know if you'd like me to elaborate further! Here are 5 related questions you might find useful:

  1. What are the primary applications of Boolean algebra in digital circuits?
  2. Can Boolean algebra be used to simplify a real-world logical expression?
  3. What are De Morgan's laws, and how do they apply in Boolean algebra?
  4. How can Boolean expressions be simplified using a Karnaugh map?
  5. What are the differences between Boolean algebra and standard algebra?

Tip: Always look for laws like Distributive, Idempot

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Math Problem Analysis

Mathematical Concepts

Boolean Algebra
Logic
Mathematical Proofs

Formulas

a.(a+b) = a
(a.b).c = a.(b.c) = (a.c).(b.c)

Theorems

Distributive Law
Idempotent Law
Absorption Law
Associative Law
Commutative Law

Suitable Grade Level

College Level