How to prove expressions using field properties?

Question

I'm in this Algebra course in which they are asking me to prove the following rules using field properties:

$$\bullet\,(a+c)+(d+b) = (a+b) + (c+d)$$

And 3 more, but I really don't know where to start. Thanks!

Answer 1

By comment of Lord Shark the Unknown, we have \begin{align*}(a+c)+(d+b)&=(c+a)+(d+b)\\&=c+(a+(d+b))=c+((d+b)+a)= c+(d+(b+a))\\&=(c+d)+(b+a)=(a+b)+(c+d)\end{align*}