CH 1 Naive Set Theory Flashcards
State Russell’s Paradox.
State Leibniz’s Principle.
Define an empty set.
State and prove uniqueness of the empty set.
Define a family of sets.
Define the union and intersection of a family of sets.
What happens to the family, union and intersection if the index set is empty?
Define the power set (via defining a subset).
Define an ordered pair using only sets.
State and prove a proposition on two ordered pairs being equal.
Define a cartesian product.
Define a function.
Define injectivity, surjectivity and bijectivity.
State and prove a theorem on functions from sets to their power sets, which implies theres no largest set.
Define the cartesian product of multiple sets, by re-interpreting the definition for two sets.
What about the cartesian products of infinite families - what if Si is the empty set for some i?
Define a binary relation.
Define 5 types of binary relation.
Define an equivalence relation.
Given an equivalence relation R, what does R(x) denote?
Define a partial order.
Define a total order.
Construct the natural numbers.
Define addition and multiplication of the natural numbers (by induction).
Why do all other number systems arise?
Construct the integers.
Define addition and multiplication in these integers.
Check that these operations are well-defined and that every equation has a solution in this system.
Repeat to construct the rationals from the integers.
Construct the reals from the rationals via Cauchy sequences.
Construct the complex numbers (and their operations).
