Front: What are the axioms of set theory?
Back: Axiom of extension, Axiom of specification, Axiom of pairing, Axiom of unions, Axiom of powers, Axiom of infinity, Axiom of choice, Axiom of substitution

Front: How does an injective, non-surjective function look like?

Front: How does a surjective non-injective function look like?

Front: How does a non-injective non-surjective function look like?

Front: Synonyms for injective
Back: one-to-one

Front: Synonyms for surjective
Back: "maps onto"

Front: What is the definition of family?
Back: A family is a function from an index set into an indexed set \(f: I \rightarrow X\)

Front: What does the phrase "a family \(\{A_i\}\) of subsets of \(X\)" refer to?
Back: A function \(A\) from some set of indices \(I\) into \(\mathcal{P}(X)\)

Front: What is \(\{a\}\) called?
Back: A "singleton" of a

Front: For a function \(f: X \rightarrow Y\), what do you call \(X\)?
Back: The domain, dom \(f=X\).

Front: For a function \(f: X \rightarrow Y\), what do you call Y?
Back: The range, ran \(f=Y\)

Front: State the axiom of extension.
Back: Two sets are equal if and only if they have the same elements.

Front: State the axiom of specification.
Back: To every set \(A\) and to every condition \(S(x)\) there corresponds a set \(B\) whose elements are exactly those elements \(x\) of \(A\) for which \(S(x)\) holds.

Front: State the axiom of pairing.
Back: For any two sets there exists a set that they both belong to.

Front: State the axiom of unions.
Back: For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.

Front: State the axiom of powers.
Back: For each set there exists a collection of sets that contains among its elements all the subsets of the given set.

Front: For a set \(x\) that represents a number, how do we define the successor \(x^+\) of \(x\)?
Back: \(x^+=x \cup \{x\}\)

Front: How is the number 0 defined?
Back: As the empty set: \(0=\emptyset\)

Front: State the axiom of infinity.
Back: There exists a set containing 0 and containing the successor of each of its elements.

Front: When is a set \(A\) the successor set?
Back: If \(0 \in A\) and \(x \in A \Rightarrow x^+ \in A\).

Front: What is a sequence?
Back: A family \(\{x_i\}\) whose index set is either a natural number (finite sequence) or the set of all natural numbers (infinite sequence).

Front: State the first Peano axiom.
Back: \(0 \in ω\) for \(0=\emptyset\) and \(ω\) being the set of all natural numbers (the unique successor set that is the subset of every successor set).

Front: State the second Peano axiom.
Back: If \(n \in ω\), then \(n^+ \in ω\).

Front: State the third Peano axiom.
Back: If \(S \subseteq ω\) and \(0 \in S\) and if \(n \in S \Rightarrow n^+ \in S\), then \(S=ω\).
This states the minimality property of \(ω\).

Front: State the fourth Peano axiom.
Back: \(n^+ \not=0\) for all \(n\) in \(ω\).

Front: State the fifth Peano axiom.
Back: If \(n, m \in ω\) and if \(n^+=m^+\), then \(n=m\).

Front: State the recursion theorem.
Back: If a is an element of a set \(X\), and if \(f\) is a function from \(X\) into \(X\), then there exists a function \(u\) from \(ω\) into \(X\) such that \(u(0)=a\) and such that \(u(n^+)=f(u(n))\) for all \(n\) in \(ω\).

Front: What is the set relation of \(m\) and \(n\) if we can write \(m<n\)?
Back: \(m \subset n\)

Q: Two sets \(E,F\) are called [...], also written [...], if [...].
A: equivalent; \(E \sim F\); there is a bjiection between them

A set \(E\) is called finite iff it is equivalent to some natural number, and infinite otherwise.

Q: A relation \(R\) in a set \(X\) is called [...] iff [...].
A: antisymmetric; for every \(x, y \in X\), \(xRy\) and \(yRx\) implies that \(x=y\)

Q: A [...] on a set \(X\) is a [...] relation on \(X\), usually written as [...].
A: partial order; reflexive, antisymmetric and transitive; \(\le\)

Q: A partial order that also satisfies [...] is called [...].
A: completeness; a total (or simple, or linear) order

Q: A [...] is called a [...].
A: totally ordered set; chain

Q: Given any relation \(R\) in a set \(X\), we can define the relation \(S\) in \(X\) by defining [...].
A: \(xSy\) iff \(xRy\) and \(x \not =y\)

Given a partially ordered set \(X\) with an element \(a \in X\), the initial segment of \(a\), written \(s(a)\), is defined as the set of all elements in \(X\) that are strictly smaller than \(a\), mathematically \(\{x \in X: x<a\}\).

Given a partially ordered set \(X\) with an element \(a \in X\), the weak initial segment of \(a\), written \(\bar{s}(a)\), is defined as the set of all elements in \(X\) that are smaller than or equal to \(a\), mathematically \(\{x \in X: x \le a\}\).

Q: An element \(a \in X\) of a partially ordered set \(X\) is [...] of \(X\) iff [...].
A: the least (smallest, first) element; \(a \le x\) for every \(x \in X\)

Q: An element \(a \in X\) of a partially ordered set \(X\) is [...] of \(X\) iff [...].
A: the greatest (largest, last) element; \(a \ge x\) for every \(x \in X\)

Q: An element \(a \in X\) of a partially ordered set \(X\) is [...] of \(X\) iff [...].
A: a minimal element; there is no \(x\) so that \(x < a\)

Q: An element \(a \in X\) of a partially ordered set \(X\) is [...] of \(X\) iff [...].
A: a maximal element; there is no \(x\) so that \(a < x\)

Q: An element \(a \in X\) of a partially ordered set \(X\) is [...] of \(E \subseteq X\) iff [...].
A: a lower bound; \(a \le x\) for all \(x \in E\)

Q: An element \(a \in X\) of a partially ordered set \(X\) is [...] of \(E \subseteq X\) iff [...].
A: an upper bound; \(a \ge x\) for all \(x \in E\)

Q: Given a partially ordered set \((X, R)\) and a subset \(E \subseteq X\), the [...] is written [...].
A: set of lower bounds on \(E\); \(E_*\)

Q: Given a partially ordered set \((X, R)\) and a subset \(E \subseteq X\), the [...] is written [...].
A: set of upper bounds on \(E\); \(E^*\)

Q: The [...] is also called the [...].
A: unique greatest element of a set of lower bounds \(E_*\); infimum

Q: The [...] is also called the [...].
A: unique least element of a set of upper bounds \(E^*\); supremum

Front: State the axiom of choice.
Back: The Cartesian product of a non-empty family of non-empty sets is non-empty.

Q: [...] states that if [...], then [...].
A: Zorn's lemma; \(X\) is a partially ordered set such that every chain in \(X\) has an upper bound; \(X\) contains a maximal element