Assume that each variable in the network is Boolean. We have studied conditional independence properties in Bayes Nets by investigating triples and finding out active paths to conclude that the independence between variables is not guaranteed given evidence at other variables. The proof that d$-$separated nodes are conditionally independent can be complicated.$$

Whether there is Gas in the car and whether the car Radio plays are independent given evidence about whether the SparkPlugs fire.$$

Gas and Radio are independent if it is known if the Battery works.$$

Gas and Radio are independent given no evidence at all. But they are dependent given evidence about whether the car Starts. For example, if the car does not start, then the radio playing is increased evidence that we are out of gas. Gas and Radio are also dependent given evidence about whether the car Moves, because that is enabled by the car starting.