# Beginner proofs in Serge Lang's Basic Mathematics

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).

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 $x + (y + z)$.
$$S = (a+b) + (d + c)\\ = a +(b+ (d+c)) \ \ \text{associativity on a, b, (d + c)}\\ = a + ((b + d) + c) \ \ \text{associativity on b, d, c}\\ = a + ((d + b) + c) \ \ \text{commutativity on b, d} \\ = a + (d + (b + c)) \ \ \text{associativity on d, b, c} \\ = (a + d) + (b + c) \ \ \text{associativity on a, d, (b + c)} \\$$
• 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)\$. – Siddharth Bhat Aug 9 '16 at 15:41