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.

# Math 3000 Abstract Math Jeopardy

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