Data science & ML Video Tutorials Part II - Group & Set Theory (Groups, Rings & Fields)

We continue building the knowledge required for basic probability theory and statistics (needed for data science & ML).

The videos in this post are about set algebra and algebraic structures like groups, rings & fields. It includes an introduction to identity, inverse and idempotent elements. We'll see that Boolean conjunction and disjunction can be seen as multiplication and addition and investigate possible additive and multiplicative identities and inverses. This is followed by power sets and basic operations on elements of power sets (union, intersection, complement, difference). All concepts are then combined in a discussion of algebraic structures including groups, rings and fields. We end with Boolean rings and describe set algebra as an example Boolean ring. In the review session we complete an example proof from CH1 of W. Rudin's Principles Of Mathematical Analysis using the axioms of a field.

Introduction to part II. The main subject of the sessions of Part II is algebraic structures. Identity and inverse elements of sets, closure of math operations, groups, rings & fields


Part II - S5. Boolean addition & multiplication. View the Boolean OR, AND operations as addition and multiplication, respectively.


Part II - S6. Additive and multiplicative identity & inverse elements. Description of special elements of a set on which an operation is defined called identity and inverse elements. We then look at additive and multiplicative elements of addition and multiplication operations (if they exist). We end up with some axioms associated with identity and inverse elements. These concepts are needed in the discussion of algebraic structures


Part II - S7. Idempotent elements & functions. Idempotent elements, functions & operations. Boolean idempotents. Axioms arising from idempotence. Both main Boolean binary operations (AND, OR) are idempotent while negation is not


Part II - S8. Boolean polynomials. Review polynomial. See that because of idempotence, the integer exponent on a Boolean variable vanishes. Coefficients vanish as well in Boolean polynomials. Every Boolean expression is equivalent to a Boolean polynomial.


Part II - S9. Sets of sets, powerset. Sets of sets (nested sets) like the powerset. Cardinality of sets of sets


Part II - 10. Set union, intersection operations and their properties. Basic set operations like the intersection and union. Properties of the set union and intersection. Example operations on the elements of the powerset of the Boolean domain B. See how the set union and intersection are related to logical OR, AND resp.


Part II - 11. Identity and inverse (if possible) elements of set addition and set multiplication. Uncover identity elements for set addition (set union) and set multiplication (intersection) operations. Explore possibility of inverse elements for these operations.


Part II - 12. The set difference. The set difference X - Y. Relative and absolute difference. And the set difference in terms of intersection with complement set. Comparison to the difference of Boolean numbers and real numbers.


Part II - 13. Algebraic structures: (Abelian) groups, rings & fields. An algebraic structure is a set along with at least one mathematical operation that is closed on all elements of that set. We look at the simplest example: a magma. From there we move to semigroups, monoids, groups, Abelian groups, rings and fields.


Part II - 14. Set algebra is a Boolean ring with symmetric difference and intersection. We review algebraic structures called rings. We look at modular addition and congruence. Then we see set algebra is a ring comprised of the power set along with the set symmetric difference and intersection operations.


Part II - Review QA (sessions 5 - 14). Part II Review Q&A. This review includes the proof using the axioms of a field for an example from W. Rudin's principles of mathematical analysis.