Logic & Proofs
Sets &
Relations
Functions & Cardinality
Groups
Analysis
100
This is a proof technique where you start by assuming the negation of what you want to prove.
What is a proof by contradiction?
100
All mathematical objects can be built from this set.
What is the empty set?
100
This is the number of bijections from a set with n elements to itself.
What is n! ?
100
This is the number of cosets of 4Z in Z.
What is 4?
100
This is usually the first sentence of a convergence proof.
What is "Let epsilon > 0." ?
200
You can disprove a "for all" statement by finding one of these.
What is a counterexample?
200
This is a subset of the Cartesian product of two sets.
What is a relation?
200
This type of set is in bijection with the set of natural numbers.
What is a countably infinite set?
200
This theorem says that the cardinality of a finite group is equal to the product of the cardinalities of a subgroup and the number of its cosets.
What is Lagrange's Theorem?
200
This Principle is true for natural numbers but not for real numbers.
What is the Well-Ordering Principle?
300
They tell you how to negate conjunctions and disjunctions.
What are De Morgan's Laws?
300
This proof technique is valid because of the Peano Axioms for natural numbers.
What is a proof by induction?
300
The inverse relation of this type of function is also a function.
What is a bijective function?
300
This is the smallest non-abelian group.
What is S_3?
300
This is a function from natural numbers to real numbers.
What is a sequence?
400
This is formed by switching the hypothesis and conclusion of a conditional statement.
What is a converse?
400
These are the three properties of an equivalence relation.
What are reflexivity, symmetry, and transitivity?
400
The (0,1) interval has the same cardinality as -------- of natural numbers.
What is the power set?
400
This is the number of subgroups of Z_6 (under addition).
What is 4?
400
For any real numbers x and y, |x + y| <= |x| + |y|.
What is the triangle inequality?
500
This logical statement is formed by negating both the hypothesis and the conclusion, and then interchanging their places.
What is a contrapositive?
500
The set of equivalence relations on a set is in bijection with the set of ------- of the same set.
What are partitions?
500
There is no --------- function from a set to its own power set.
What is surjective? (or bijective)
500
This is the inverse of [4] in U(7), which is {1,2,3,4,5,6} under multiplication modulo 7.
What is 2?
500
This is usually the last sentence of a divergence proof.
What is "|s_n - s| > epsilon"?
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