Definition of Turing machines and rejection states In some definitions of Turing machine, there is only a set of accepting states and no mention of a set of rejecting states. 

But it seems to me that the definition that includes only a set of accepting states and that includes also a set of rejecting states will differ.

How can they actually be shown to be equivalent? Or is it not the case?
A deterministic Turing machine accepts if, and only if, it terminates in an accepting state. Any state that is not accepting is rejecting by default.  There are various equivalent ways of defining Turing machines: for example, instead of having a HALT instruction and accepting and rejecting states, you could have explicit ACCEPT and REJECT instructions.