This is a long abstract ... no doubt this is because if I shortened it the topic might possibly seem too trivial to warrant a full seminar to discuss it.
The topics: Set Theory, Propositional Logic (without quantifiers), Digital Logic (in Computing/Electrical Engineering) and Boolean Algebra are all essentially the same thing ... though we tend to use different notation and there are different emphases*.
In Set Theory we meet the Venn diagram - we have all drawn 3 intersecting circles in such a way that the universal set is partitioned into 8 regions - this represents the most general way that three sets can intersect (some regions may of course be empty). The significance of 8 as a power of 2 we tend to explore in Boolean Algebra - the cardinalities of Boolean Algebras are necessarily powers of 2.
A Carroll diagram has a similar function to a Venn diagram ... the universal set U is drawn as a square or rectangle and we use straight lines rather than circles to partition U ... which works nicely for representing the most general possibility with 0, 1 and 2 sets. At first we appear to lose contiguity (one-pieced-ness) when representing the general situation with 3 sets ... but if we identify a pair of opposite sides (so that the rectangle representing U is topologically equivalent to a cylinder) we can restore contiguity. We can maintain contiguity with 4, 5 and even 6 sets with similar topological tricks ... after that it becomes less than practical.
The advantage of a Carroll diagram over a Venn diagram is that the regions are regular. What does one do to represent the general situation with 4 or more sets for a Venn diagram? ... there is a solution that maintains contiguity but at most 3 of the intersecting shapes is a circle ... the solution is not particularly nice.
Another advantage of the Carroll diagram is that the Boolean Algebra structure is apparent in the diagram - subalgebras (I really mean sub-universal sets) also have regular shapes. It is this last property that was exploited in 1953 by M. Karnaugh, who used it to simplify digital circuitry ... and so Carroll diagrams became known as Karnaugh maps.
I will give an example to show how Engineers use the Carroll diagram/Karnaugh map ... and discuss how they are useful for mathematicians in sorting out the logic of computer algorithms and ensuring one has covered all possibilities in searches involving multiple conditions in an efficient way. It's also useful in statistics - joint probability tables double as Carroll diagrams!
I should also have mentioned that Carroll diagrams are named for Lewis Carroll (Charles Dodgson) who was a logician as well as a famous children's book writer.
* - plural of emphasis ... I checked, it is a word.