1 Naive Set Theory Flashcards
Define an empty set and prove it is unique by Leibniz’s Principle.
Define the union and intersection of families of sets.
What happens if the index set is empty?
An empty set is a set without any members.
Leibniz’s Principle states that two sets are
equal if and only if they have exactly the same members.
Proof: Suppose that E1 and E2 are empty sets. Then they have exactly the same
members (namely, none at all!), and so by Leibniz’s Principle, they are equal.
Define the power set of B.
Define an ordered pair (x,y).
State proposition: When are 2 ordered pairs equal? Prove the definition or odrered pairs from above holds.
Proposition 1.2 (x, y) = (u, v) if and only if x = u and y = v.
Proof “If” is clear; we prove “only if”. First note that {{x},{x, y}} is a set of two elements if x ̸= y, or one element if x = y. So, assuming that (x, y) = (u, v), we have two cases:
* If this set has only one element, then x = y = u = v.
* If it has two elements, then one of them ({x} or {u}) is a 1-element set and the other is a 2-element set. So the 1-element sets must be equal, whence x = u; and the 2-element sets are also equal, whence y = v.
Define the Cartesian Product of two sets.
Define a function.
Define injectivity, surjectivity and bijectiivity of a function.
State and prove Thm on injective/ surjective functions from A –> P(A).
Define the cartesian product of more than 2 sets.
What if Si is the empty set for some i?
What about the converse? State the Axiom of Choice.
Define:
- Binary Relation
- 5 properties of a binary relation
- Equivalence Relation
- Partial Order
- Total Order
Construct the natural numbers.
Show this implies proof by induction holds.
Define addition and multiplication of the natural numbers.
Construct the integers from the natural numbers.
Construct the rationals from the integers.
Construct the reals from the rationals.
Construct the complex numbers from the reals.
