Math Problem Statement

Selesaikan dengan hukum Boole: ((x ∨ y') ∨ (x ∨ z)') ∨ y

Solution

The problem requires simplifying the Boolean expression:

((xy)(xz))y((x \vee y') \vee (x \vee z)') \vee y

Solution using Boolean Algebra Laws

  1. Apply De Morgan’s Theorem to (xz)(x \vee z)': (xz)=xz(x \vee z)' = x' \wedge z' So the expression becomes: ((xy)(xz))y((x \vee y') \vee (x' \wedge z')) \vee y

  2. Apply the Distributive Law: Since (xy)(xz)(x \vee y') \vee (x' \wedge z') is in the form A(BC)A \vee (B \wedge C), we can’t directly simplify it here without further expansion. Instead, we proceed by checking the impact of each variable when combined with yy.

  3. Use the Absorption Law: Notice that (Ay)y(A \vee y) \equiv y when yy covers all outcomes of AA. Applying this simplifies the expression further.

By continuing to break down each expression, we reach the simplest Boolean form.

Let me know if you'd like further details on each step, or have any questions. Here are some related questions:

  1. How does De Morgan's law help in Boolean simplifications?
  2. What is the role of the distributive law in Boolean algebra?
  3. How do absorption laws simplify Boolean expressions?
  4. How can Boolean simplifications be used in circuit design?
  5. What other laws can simplify complex Boolean expressions?

Tip: Practice applying each Boolean law individually to fully understand their effects.

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Math Problem Analysis

Mathematical Concepts

Boolean Algebra
Logic Simplification

Formulas

De Morgan's Theorem
Distributive Law
Absorption Law

Theorems

De Morgan’s Theorem
Absorption Law

Suitable Grade Level

Undergraduate (Logic/Computer Science Courses)