Based on a model of the random experiments and their resulting events as sets, elements and subsets with certain measure allocated to them, we can introduce logical operations on events and express them through logical operations on sets and subsets.
Let's analyze an example. Assume, we are interested in a single dice rolling of either a number divisible by 2 or divisible by 3. There are 6 elementary events with a probability of each 1/6 in this experiment. Four results 2, 3, 4, 6 are the event we are interested, so the probability of this event should be 4/6=2/3. Let's approach this from the more formal viewpoint. The results of an experiment are modeled as a set of 6 elements {1,2,3,4,5,6} with a measure of 1/6 allocated to each one. The event "Result is divisible by 2" is represented as a subset {2,4,6} (three elements). The event "Result is divisible by 3" is represented as a subset {3,6} (two elements). As you see, these two subsets have a common element {6}. Simple addition of measures to obtain a probability of occurring one OR another event results in the wrong answer 5/6. But that is because a measure of a union of two subsets is not equal to a sum of their measures in case there are common elements. First, to model an OR condition between two events we should perform a union of subsets that represent these events according to the rules of set operations. This union is {2,3,4,6} (we included {6} only once as a union operation requires). Now a new subset, which is a union of two subsets, has a measure of 4/6=2/3, as it should.
So, as we see, a representation of an OR logical operation between two events can correctly be represented in a formal model as a true union between subsets representing these two events.
Let's examine other logical operations between events. The operation AND is naturally represented as an intersection between subsets that represent our events. Consider the following example. We are interested in getting a number divisible by both 2 and 3, when rolling a single dice. Obviously, there is only one outcome that satisfies this condition - number 6. So, the probability of the occurrence of this event equals to 1/6. From the formal standpoint of the set theory, the event "Result is divisible by 2" is represented as a subset {2,4,6} (three elements), the event "Result is divisible by 3" is represented as a subset {3,6} (two elements). The intersection of these two subsets is an element {6}, that represents an event "Result is divisible by 2" AND "Result is divisible by 3".
The last logical operation we consider is NOT. Its formal representation in the set theory is an operation of complement. A single dice rolling event "Result is not divisible by 3" is comprised from elementary events 1, 2, 4 and 5. Formally, this event is represented as a complement to a representation of an event "Result is divisible by 3", which is a subset {3,6} in a set representing our sample space.
Here is the final thought we can derive from the above analysis of the correspondence between logical operations on random events and the set theory operations on representations of these events as subsets of some set describing the sample space. An entire Theory of Probabilities can be considered as a particular case of the set theory with an additive measure introduced on subsets of a main set that represents the sample space of a random experiment with a condition that the measure of an entire set equals to 1.