Math Problem Statement

Show that (π‘₯ + 𝑦) + 𝑧 = π‘₯ + (𝑦 + 𝑧) for all π‘₯, 𝑦, 𝑧 ∈ 𝐅𝑛

Solution

To prove that (x+y)+z=x+(y+z)(x + y) + z = x + (y + z) for all x,y,z∈Fnx, y, z \in \mathbb{F}_n, we need to show that the operation of addition in the finite field Fn\mathbb{F}_n is associative.

Step 1: Define the Elements

Let x,y,zx, y, z be elements of the finite field Fn\mathbb{F}_n. In Fn\mathbb{F}_n, addition is defined modulo nn. Therefore, for any two elements a,b∈Fna, b \in \mathbb{F}_n, the addition a+ba + b is given by the usual integer addition followed by taking the result modulo nn.

Step 2: Apply Addition Modulo nn

We want to prove the associative property of addition:

(x+y)+z=x+(y+z)(x + y) + z = x + (y + z)

Let's consider both sides of this equation.

Left-Hand Side:

(x+y)+z=[(x+y)mod  n]+z(x + y) + z = [(x + y) \mod n] + z Now, add zz to the result: =[(x+y)mod  n+z]mod  n= [(x + y) \mod n + z] \mod n

Right-Hand Side:

x+(y+z)=x+[(y+z)mod  n]x + (y + z) = x + [(y + z) \mod n] Now, add xx to the result: =[x+(y+z)mod  n]mod  n= [x + (y + z) \mod n] \mod n

Step 3: Use Properties of Modular Arithmetic

In modular arithmetic, addition is associative. Therefore: [ [(x + y) \mod n + z] \mod n = [x + (y + z) \mod n] \mod n ] This simplifies to: [ [(x + y + z) \mod n] = [(x + y + z) \mod n] ]

Step 4: Conclusion

Since both sides are equivalent, we have shown that: (x+y)+z=x+(y+z)(x + y) + z = x + (y + z) for all x,y,z∈Fnx, y, z \in \mathbb{F}_n.

Summary

The operation of addition in the finite field Fn\mathbb{F}_n is associative, and thus (x+y)+z=x+(y+z)(x + y) + z = x + (y + z) holds true for any elements x,y,zx, y, z in Fn\mathbb{F}_n.

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Related Questions:

  1. What is the definition of a finite field Fn\mathbb{F}_n?
  2. How does modular arithmetic work in Fn\mathbb{F}_n?
  3. What are the properties of addition in a finite field?
  4. How do you prove the commutative property of addition in Fn\mathbb{F}_n?
  5. Can you demonstrate the distributive property in Fn\mathbb{F}_n?

Tip: Associativity is a fundamental property of binary operations, and it's essential for the structure of fields and rings in abstract algebra.

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Math Problem Analysis

Mathematical Concepts

Associativity
Finite Fields
Modular Arithmetic

Formulas

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Theorems

Associativity of Addition

Suitable Grade Level

Undergraduate Level