Sets & Cardinalities  Mathematical Logic  Relations & Functions  Proof Techniques  Results & Conjectures 

100
What is the empty (null/void) set?
The only set with zero cardinality.

100
What is a statement?
A declarative sentence that is either true or false.

100
What is bijective?
A relation and its inverse are functions if and only if they satisfy this property.

100
What is a counterexample?
To disprove a statement, it suffices to give this.

100
What is a lemma?
A mathematical result that is used as an auxiliary result, often to prove a theorem.

200
What is the set of rational numbers?
Every element in this set can be written as the quotient of an integer and a natural number.

200
What it is a disjunction?
The combination of two statements using a logical OR.

200
What is an equivalence relation?
A relation that is reflexive, symmetric, and transitive.

200
What is QED (quod erat demonstrandum)?
The English translation of this Latin abbreviation is "what had to be shown".

200
What is the Fundamental Theorem of Arithmetic?
Every integer greater than 1 can be written as a unique product of prime numbers.

300
What are pairwise disjoint sets?
A collection of nonempty subsets of a set A is called a partition of A if its elements contain all elements of A and satisfy this property.

300
What is the universal quantifier (∀)?
This quantifier means FOR ALL.

300
What is surjective (onto)?
If a function's range equals its codomain, then it has this property.

300
What is a proof by contradiction?
To show that P ⇒ Q, suppose that P is true and Q is false.

300
What is Goldbach's Conjecture?
Every even integer greater than 2 can be written as the sum of two prime numbers.

400
What are the integers modulo n?
Given a positive integer n, this set contains all equivalence classes of integers that have the same remainder upon division by n.

400
What is a necessary condition?
A requirement without which a conclusion is always false.

400
What is the identity?
The composition of a bijective function with its own inverse.

400
What is without loss of generality (wlog)?
A commmon justification to shorten a proof if different cases are identical or very similar.

400
What is Euclid's Lemma?
Let m and n be integers, and p be a prime number: p  mn ⇒ p  m ∨ p  n

500
What is aleph null?
The cardinal number of each denumerable set.

500
What is the Modus Ponens?
P ∧ (P ⇒ Q) ⇒ Q

500
What is a dictionary (alphabetical/lexicographic) order
? eight five four nine one seven six three two
When sorting the numbers from 1 to 9, 1 < 2 but 2 > 3 > 4 > 5, 5 < 6 but 6 > 7 > 8, 8 < 9 but 9 < 1. 
500
What is the Principle of Mathematical Induction?
( ∀ n ≥ n_{0}: P(n) ) ≡ ( P(n_{0}) ∧ ∀ k ≥ n_{0}: P(k) ⇒ P(k+1) )

500
What is the SchröderBernstein Theorem?
If the cardinal number of a set A is both lessthanorequal and greaterthanorequal than the cardinal number of a set B, then A and B are numerically equivalent.

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