For $(a,b), (c,d)\in \mathbb{R}^2$ define $(a,b)\sim (c,d)$ to mean that $2a−b = 2c−d$. Prove that $\sim$ is an equivalence relation on $\mathbb{R}^2$.
Reflexive: let $a$ be an element in $\mathbb{R}$. so $2a - a = 2a - a$. therefore $a\sim a$ is reflexive.
Symmetric: let $a,b,c,d$ be elements in $\mathbb{R}$. so $2a - b = 2d - c$. Since $2d - c = 2a - b$ it is symmetric.
Is the symmetric part right? would i just say that $(a,c)\sim (d,c)$ and $(d,c)\sim (a,b)$?what about for transitive?
The relation $\sim$ is defined on $\mathbb{R}^2$ so to prove it is reflexive you need to take an element $(a,b)\in \mathbb{R}^2$ and show that $(a,b)\sim (a,b)$. Also, the fact that $\sim$ is symmetric and transitive follows from the fact that $=$ is symmetric and transitive.