Fundamentals of Formalisation Level 3: Set Theoretic Relations and Enumerability

Followup to Fundamentals of Formalisation level 2: Basic Set Theory. The big ideas:

• Ordered Pairs

• Relations

• Functions

• Enumerability

• Diagonalization To move to the next level you need to be able to:

• Define functions in terms of relations, relations in terms of ordered pairs, and ordered pairs in terms of sets.

• Define what a one-to-one (or injective) and onto (or surjective) function is. A function that is both is called a one-to-one correspondence (or bijective).

• Prove a function is one-to-one and/​or onto.

• Explain the difference between an enumerable and a non-enumerable set. Why this is important:

• Establishing that a function is one-to-one and/​or onto will be important in a myriad of circumstances, including proofs that two sets are of the same size, and is needed in establishing (most) isomorphisms.

• Diagonalization is often used to prove non-enumerability of a set and also it sketches out the boundaries of what is logically possible. You can find the lesson in our ihatestatistics course. Good luck! P.S. From now on I will posting these announcements instead of Toon Alfrink.