I'm a re-igniting my passion for maths and recovering from math anxiety. I'm using Serge Lang's Basic Mathematics to re-learn useful high school topics before attempting more difficult subjects. I'm new to proofs as most of what I did before was computational.

The first few exercises ask me to justify each step using commutativity and associative to in proving the following identities:

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

I could use an example of how to go about this. Thank you.

  • $\begingroup$ If there is an answer which you consider good enough, I strongly advise you to accept it (by clicking on the "check" mark below the vote number of the answer), and to upvote it, if you believe it is fair. See math.stackexchange.com/help/accepted-answer $\endgroup$ – Luiz Cordeiro Aug 9 '16 at 15:55

Let $$ S = (a+b)+(c+d) $$ We need to show that $$ S =(a+d)+(b+c) $$

The basic idea is to use commutativity to swap, say $(x + y)$ to $(y + x)$ and to use associativity to re-organize brackets as we like, to convert something like $(x + y) + z$ to $x + (y + z)$.

$$ S = (a+b) + (d + c)\\ = a +(b+ (d+c)) \ \ \text{associativity on a, b, (d + c)}\\ = a + ((b + d) + c) \ \ \text{associativity on b, d, c}\\ = a + ((d + b) + c) \ \ \text{commutativity on b, d} \\ = a + (d + (b + c)) \ \ \text{associativity on d, b, c} \\ = (a + d) + (b + c) \ \ \text{associativity on a, d, (b + c)} \\ $$

  • $\begingroup$ Much obliged. Am I correct in assuming that this is the manipulation of algebraic expressions to show equality? That this is an exercise in manipulating these equations (using the principles) in as many ways as I can? $\endgroup$ – Jebussy Aug 9 '16 at 15:39
  • $\begingroup$ Yes, you're doing this to show that the left hand right is the same as the right hand side. Equality is what you're looking for between $(a + d) + (b + c)$ and (a + b) + (d + c)$. $\endgroup$ – Siddharth Bhat Aug 9 '16 at 15:41

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