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 ≥ n0: P(n) ) ≡ ( P(n0) ∧ ∀ k ≥ n0: P(k) ⇒ P(k+1) )

500

What is the Schröder-Bernstein Theorem?
If the cardinal number of a set A is both less-than-or-equal and greater-than-or-equal than the cardinal number of a set B, then A and B are numerically equivalent.

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