MHB How to Justify Each Step Using Commutativity and Associativity?

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The discussion focuses on verifying the expression (a-b)+(c-d) = (a+c)+(-b-d) using the properties of associativity and commutativity. The user demonstrates each step of the transformation, applying associativity to regroup terms and commutativity to rearrange them. The final expression confirms that the original and transformed equations are equivalent. The verification process emphasizes the importance of these mathematical properties in simplifying expressions. Overall, the approach effectively illustrates how to justify each step in the equation.
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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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Looks good to me.
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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