The video explains why the statements "all unicorns are purple" and "all unicorns can do math" are true in mathematics, using mathematical logic and the concept of quantifiers. The video recommends the book "How to Prove It: A Structured Approach" by Velleman as a great resource for learning how to write proofs. The book's content includes logic, proofs, relations, functions, mathematical induction, and infinite sets. The book has numerous exercises, but no solutions provided. The video demonstrates the quantifier negation law and how it applies to empty sets.
Facts:
1. All unicorns are purple and all unicorns can do math are true in mathematics.
2. "How to Prove It: A Structured Approach" is a book that explains why those statements are true.
3. The book is written by Velman and is good for anyone who wants to learn how to write proofs.
4. The book has exercises but no solutions are provided.
5. The book covers logic, proofs, relations, functions, mathematical induction, and infinite sets.
6. The author breaks down the scratch work in proofs and shows how to figure out the proofs in detail.
7. The double negation law is an important logic law used in calculus proofs and it is described in the book.
8. The statement "for all x in a, p of x" is true if a is the empty set.
9. The book is recommended to anyone who wants to learn more about mathematical logic and proof writing.