Induction
A least element of A.
(Let Ø ≠ A ⊆ ℝ.) What is a number m ∈ A if m ≤ x for all x ∈ A?
A relation R on A if aRa for all a in A.
What is a reflexive relation?
f(A1 ∩ A2) = f(A1) ∩ f(A2) (T/F)
False. f(A1 ∩ A2) ⊆ f(A1) ∩ f(A2
A is infinite.
What is A if it is not finite?
This proves the set ℚ+ of positive rational numbers is countable.
1/1 1/2 1/3 1/4....
2/1 2/2 2/3 2/4...
3/1 3/2 3/3 3/4...
4/1 4/2 4/3 4/4...
....
It is well-ordered.
What is a nonempty subset A ⊆ ℝ if every nonempty subset of A has a least element?
An equivalence relation.
What is a relation that is reflexive, symmetric, and transitive?
The inverse of f.
The function f-1: B→A defined by f-1(b) = a if any only if f(a) = b is what?
(The pigeonhole principle) There is no injective function from A to B.
What if A and B are finite and |B| < |A|?
The contradiction in the proof for Euclid's Theorem.
What is "if there are a finite number of primes 'a', then construct a number (1 x 2 x 3 x 5...x a) + 1 = k, then k must be both prime and composite"?
The cardinality of the set is 2n.
What is the cardinality of the power set?
The two sets are disjoint.
Sets A and B are what, if A n B=∅?
If g○f is bijective.
When is f injective when g is surjective?
That's why the proof for |ℝ| = Ĉ doesn't work for ℚ.
What is important about the fact that b is irrational in the proof for "(1,0) is uncountable"?
The Division Algorithm. (The Theorem)
What is "let a, m in ℤ. Then there exist unique q, r in ℤ such that a = qm + r and 0 ≤ r ≤ m"?
The premises for strong induction.
What are "
Fix m ∈ ℕ and let S = {i ∈ ℤ | i ≧ m}
Let P(s) be an open sentence for which s ∈ S"?
{b in B | (a,b) in R for some a in A}
What is the range of A?
Let f: A →B and let C ⊆ A. the function f|c = C →B defined by f|c (x) = f(x).
What is the restriction of f to C.
There exists 0 ⪬ m < n such that |B| = m, and |B| ≠ n.
Let A be a set with |A| = n > 0. If B is a proper subset of A, then what?
The number created for the proof of "(0, 1) is uncountable."
What is 0.b1b2b3... where bi = {1 if i≠1, 2 if i=1?
We can conclude that P(S) is true for all s in S.
What can we conclude if:
(i) P(m) is true
(ii) ∀ k ∈ S, P(k) ⇒ P(k + 1) is true?
The family is a partition on X.
What do you call a relation that is pairwise disjoint and whose union (of all sets), is all of X?
The inclusion mapping.
What is the function I that defines A ⊆ B. define i: A→B by i(a) = a?
Prove "Theorem 7.18: |ℝ| = Ĉ" the fastest.
Check!
The contradiction in Cantor's Theorem.
What is "Define B = {a in A | a not in g(a)}. If z in B then z not in g(z) = B, If z not in B then z in g(z) = B"?