I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me?
For $a, b \in \Bbb R$, define $a \sim b$ if $ |a - b| \leq 1$. Is the relation $\sim$ reflexive? symmetric? transitive? Is $\sim$ an equivalence relation?
Thank you!
Let $a=1$, $b=2$ and $c=3$, then we have $a \sim b$ because $|1-2|=1\leq 1$ and we also have $b \sim c$ because $|2-3|=1\leq 1$. However it is not true that $a \sim c$ because $|1-3|=2\ge 1$. That is a counter example for transitivity, thus this relation is not an equivalence relation.
For the future: For absolute values, try imagining a number line in your head where you need to place two numbers and the $|a-b|\leq 1$ means that the distance between them is less than 1. From there, you can develop your argument.
By the way, this relation is symmetric: If $a \sim b$ then $|a-b|\leq 1$ but notice that $|a-b|=|(-1)(b-a)|=|-1||b-a|=|b-a|$ so $|b-a|\leq 1$ as well and $b \sim a$.