MHB How to Justify Each Step Using Commutativity and Associativity?

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SUMMARY

The discussion focuses on verifying the equation (a-b)+(c-d) = (a+c)+(-b-d) using the properties of commutativity and associativity in basic mathematics, as outlined in Serge Lang's "Basic Mathematics." The user demonstrates the step-by-step transformation of the left-hand side into the right-hand side, applying associativity and commutativity principles. The final verification confirms the correctness of the equation, showcasing the importance of these mathematical properties in simplifying expressions.

PREREQUISITES
  • Understanding of basic algebraic operations
  • Familiarity with the properties of commutativity
  • Knowledge of the properties of associativity
  • Basic experience with mathematical proofs
NEXT STEPS
  • Study the properties of commutativity in depth
  • Explore the properties of associativity in various mathematical contexts
  • Practice simplifying algebraic expressions using these properties
  • Review examples from Serge Lang's "Basic Mathematics" for additional context
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Students of mathematics, educators teaching basic algebra, and anyone looking to strengthen their understanding of algebraic properties and simplification techniques.

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Exercise 3 Chapter 1 Basic Mathematics Serge Lang

Verifying my answer.

My answer:

(a-b)+(c-d) = (a+c)+(-b-d)

Let p = (a-b)+(c-d) We need to show that p = (a+c)+(-b-d)

(a-b)+(c-d)

a+(-b+(c-d)) Associativity

a+((-b+c)-d) Associativity

a+((c-b)-d) Commutativity

((a+c)-b)-d) Associativity

(a+c)+(-b-d) Associativity
 
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