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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Sets
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{flashcard}[Definition]{Set and element}
A \emph{set} is a collection of objects; the objects in the set are called \emph{elements}.

If $A$ is a set, and $a$ is an element of $A$, then we write $a \in A$.
\end{flashcard}

\begin{flashcard}[Definition]{Cardinality of a set}
The \emph{cardinality} of a set is the number of elements in the set. 

The cardinality of the set $A$ is denoted by $|A|$ or $\#A$.
\end{flashcard}

\begin{flashcard}[Definition]{Empty set (null set)}
The \emph{empty set} is the set that contains zero elements.

The empty set is denoted by $\varnothing$.
\end{flashcard}

\begin{flashcard}[Definition]{Finite and infinite sets}
A set is \emph{finite} if its cardinality is finite, that is, the set contains finitely many elements.

A set is \emph{infinite} if it contains infinitely many elements.
\end{flashcard}

\begin{flashcard}[Definition]{Ordered pair / $n$-tuple}
An \emph{ordered pair} is an ordered list of two elements. More generally, an \emph{ordered $n$-tuple} is an ordered list of $n$ elements. The standard notation is to use a comma separated list enclosed by parenthesis: $(a_1, a_2, \dotsc, a_n)$.
\end{flashcard}

\begin{flashcard}[Definition]{Cartesian product}
If $A$ and $B$ are sets, the \emph{cartesian product} $A \times B$ is the set of ordered pairs where the first element is from $A$, and the second is from $B$.
\begin{align*}
A \times B = \{(a,b) : a \in A, b \in B\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Subset}
A set $A$ is a \emph{subset} of a set $B$ if every element of $A$ is an element of $B$.
\domino{$A \subseteq B$}{$\iff$}{if $x \in A$, then $x \in B$}
\end{flashcard}

\begin{flashcard}[Definition]{Proper subset}
The set $B$ is a \emph{proper} subset of $A$ if $B$ is a subset of $A$ that is not equal $A$, and we write $B \subset A$.
\domino{$B \subset A$}{$\iff$}{$B \subseteq A$ and $B \neq A$}
\end{flashcard}

\begin{flashcard}[Definition]{Power set}
The \emph{power set} of a set $A$ is the set of all subsets of $A$.

The power set of $A$ is denoted by $\mathscr P(A)$.
\begin{align*}
\mathscr P(A) =\{B : B \subseteq A\}
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Set equality}
Two sets, $A$ and $B$, are \emph{equal} if all the elements of $A$ are elements of $B$ and vice versa.

\domino{$A = B$}{$\iff$}{$x \in A$ if and only if $x \in B$}
\end{flashcard}

\begin{flashcard}[Definition]{Union}
The \emph{union} of two sets, $A$ and $B$, is the set of all element in $A$ or in $B$. 

The union of these sets is denoted by $A \cup B$.
\begin{align*}
A \cup B = \{x : x \in A \text{ or } x \in B\}
\end{align*}
\vspace{-4mm}
\domino{$x \in A \cup B$}{$\iff$}{$x \in A$ or $x \in B$}
\end{flashcard}

\begin{flashcard}[Definition]{Finite and infinite union}
A \emph{finite union} is the union of finitely many sets. An \emph{infinite union} is the union of infinitely many sets. 

Let $A_1, A_2, A_3, \ldots$ be sets, then
\begin{align*}
\bigcup_{i=1}^n A_i &= \{x : x \in A_i \text{ for some } 1 \le i \le n\}\\
\bigcup_{i \in \bbn} A_i &= \{x : x \in A_i \text{ for some } i \in \bbn\}
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Intersection}
The \emph{intersection} of two sets, $A$ and $B$, is the set of all element in $A$ and in $B$. 

The intersection of these sets is denoted by $A \cap B$.
\begin{align*}
A \cap B = \{x : x \in A \text{ and } x \in B\}
\end{align*}
\vspace{-4mm}
\domino{$x \in A \cap B$}{$\iff$}{$x \in A$ and $x \in B$}
\end{flashcard}

\begin{flashcard}[Definition]{Finite and infinite intersection}
A \emph{finite intersection} is the intersection of finitely many sets. An \emph{infinite intersection} is the intersection of infinitely many sets. 

Let $A_1, A_2, A_3, \ldots$ be sets, then
\begin{align*}
\bigcap_{i=1}^n A_i &= \{x : x \in A_i \text{ for all } 1 \le i \le n\}\\
\bigcap_{i \in \bbn} A_i &= \{x : x \in A_i \text{ for all } i \in \bbn\}
\end{align*}
\end{flashcard}


\begin{flashcard}[Definition]{Set difference}
If $A$ and $B$ are sets, the, is the difference $A-B$ is the set of elements in $A$ that are not in $B$.
\begin{align*}
A-B = \{x : x \in A \text{ and } x \notin B\}
\end{align*}
\vspace{-4mm}
\domino{$x \in A - B$}{$\iff$}{$x \in A$ and $x \not\in B$}
\end{flashcard}

\begin{flashcard}[Definition]{Disjoint sets}
Two sets, $A$ and $B$, are \emph{disjoint} if $A \cap B = \varnothing$.
\end{flashcard}

\begin{flashcard}[Theorem]{Double containment principle}
Let $A$ and $B$ be sets. Then $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.
\end{flashcard}

\begin{flashcard}[Definition]{Complement}
The \emph{complement} of a set $A$ is the set of all elements that are not in $A$, and is denoted by $A^c$ or $\overline A$.

If $A \subseteq B$, then the \emph{complement of $A$ in $B$} is the set of elements in $B$ that are not in $A$, i.e. $A^c = B-A$.
\begin{align*}
A^c = \{x : x \notin A\}
\end{align*}
\vspace{-4mm}
\domino{$x \in A^c$}{$\iff$}{$x \not\in A$}
\end{flashcard}

\begin{flashcard}[Theorem]{DeMorgan's laws}
For any sets $A$ and $B$, 
\begin{align*}
(A \cap B)^c &= A^c \cup B^c,\\
(A \cup B)^c &= A^c \cap B^c.
\end{align*}
\end{flashcard}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Logic
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{flashcard}[Definition]{Statement}
A \emph{statement} is a sentence or mathematical expression that is definitively true or definitively false. 
\end{flashcard}

\begin{flashcard}[Definition]{Logical statement}
A \emph{statement} is a sentence or mathematical expression that is definitely true or definitely false.
\end{flashcard}

\begin{flashcard}[Definition]{Logical and}
The statement ``\emph{$P$ and $Q$}" is true if both $P$ is true and $Q$ is true. Otherwise ``$P$ and $Q$" is false.
\end{flashcard}

\begin{flashcard}[Definition]{Logical or}
The statement ``\emph{$P$ or $Q$}" is true if $P$ is true or $Q$ is true (or both statements are true). The statement ``$P$ and $Q$" is false only if both $P$ is false and $Q$ is false.
\end{flashcard}

\begin{flashcard}[Definition]{Logical negation}
The \emph{negation} of a statement $P$ is the statement $\sim P$. The statement $\sim P$ is true if $P$ is true. The statement of $\sim P$ is false if $P$ is true.
\end{flashcard}

\begin{flashcard}[Definition]{Implies}
The statement ``\emph{$P$ implies $Q$}" ($P \implies Q$) is false if $P$ is true and $Q$ is false. Otherwise the statement is true.
\end{flashcard}

\begin{flashcard}[Definition]{Converse}
The \emph{converse} of $P \implies Q$ is the statement $Q \implies P$. In general, these two statements are independent, meaning that the truthfulness of one statement does not determine the truthfulness of the other.
\end{flashcard}

\begin{flashcard}[Definition]{If and only if}
The statement ``\emph{$P$ if and only if $Q$}" ($P \iff Q$) is equivalent to the statement $(P \implies Q) \land (Q \implies P)$. In other words, $P \iff Q$ is true if both $P \implies Q$ and $Q \implies P$ are true.
\end{flashcard}

\begin{flashcard}[Definition]{Contrapositive}
The \emph{contrapositive} of the statement ``if $P$, then $Q$" is the statement ``if $\neg Q$, then $\neg P$". These statements are equivalent, meaning that they are either both true or both false. 
\end{flashcard}

\begin{flashcard}[Definition]{Theorem}
A \emph{theorem} is a mathematical statement that is true and can be (and has been) verified as true.
\end{flashcard}

\begin{flashcard}[Definition]{Proof}
A \emph{proof} of a theorem is a written verification that shows that the theorem is definitely and unequivocally true.
\end{flashcard}

\begin{flashcard}[Definition]{Definition}
A \emph{definition} is an exact, unambiguous explanation of the meaning of a mathematical word or phrase.
\end{flashcard}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Counting
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{flashcard}[Definition]{List and entries}
A \emph{list} is an ordered sequence of objects. The objects in the list are called \emph{entries}. Unlike sets, the order of entries matters, and entries may be repeated.
\end{flashcard}

\begin{flashcard}[Definition]{List length}
The \emph{length} of a list is the number of entries in the list.
\end{flashcard}

\begin{flashcard}[Definition]{List equality}
Two lists $L$ and $M$ are \emph{equal} if they have the same length, and the $i$-th entry of $L$ is the $i$-th entry of $M$.
\end{flashcard}

\begin{flashcard}[Definition]{Empty list}
The \emph{empty list} is the list with no entries, and is denoted by $()$.
\end{flashcard}

\begin{flashcard}[Theorem]{Multiplication principle}
Suppose in making a list of length $n$ there are $a_i$ possible choices for the $i$-th entry. Then the total number of different lists that can be made in this way is $a_1 a_2 a_3\dotsb a_n$. 
\end{flashcard}

\begin{flashcard}[Definition]{Factorial}
If $n$ is a non-negative integer, then $n!$ is the number of non-repetitive lists of length $n$ that can be made from $n$ symbols. Thus $0! = 1$, and if $n > 1$, then $n!$ is the product of all integers from 1 to $n$. That is, if $n > 1$, then $n! = n(n-1)(n-2) \dotsb 2 \cdot 1$. 
\end{flashcard}

\begin{flashcard}[Definition]{$n$ choose $k$}
If $n$ and $k$ are integers, and $0 \le k \le n$, then 
\begin{align*}
\binom{n}{k} = \frac{n!}{k!(n-k)!}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Theorem]{Binomial theorem}
If $n$ is a non-negative integer, then 

\begin{align*}
(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Theorem]{Inclusion-exclusion}
If $A$ and $B$ are sets, then 

$|A \cup B| = |A| + |B| - |A \cap B|$.
\end{flashcard}

\begin{flashcard}[Theorem]{Addition principle}
If $A_1, A_2, \dotsc, A_n$ are disjoint sets, then 

$|A_1 \cup A_2 \cup \dotsb \cup A_n| = |A_1| + |A_2| + \dotsb + |A_n|$.
\end{flashcard}

\begin{flashcard}[Definition]{Even}
An integer $a$ is \emph{even} if there exists an integer $b$ such that $a = 2b$.
\domino{$a$ is even}{$\iff$}{$a = 2b$ for some $b \in \bbz$}
\end{flashcard}

\begin{flashcard}[Definition]{Odd}
An integer $a$ is \emph{odd} if there exists an integer $b$ such that $a = 2b+1$.
\domino{$a$ is odd}{$\iff$}{$a = 2b+1$ for some $b \in \bbz$}
\end{flashcard}

\begin{flashcard}[Definition]{Divides}
If $a$ and $b$ are integers, then $b$ \emph{divides} $a$ if there exists an integer $q$ such that $a = qb$. In this case, $b$ is a \emph{divisor} of $a$, and $a$ is a \emph{multiple} of $b$.
\domino{$b \mid a$}{$\iff$}{$a = qb$ for some $q\in \bbz$}
\end{flashcard}

\begin{flashcard}[Definition]{Parity}
Two integers have the \emph{same parity} if they are both even or both odd. Otherwise they have \emph{opposite parity}.
\end{flashcard}

\begin{flashcard}[Definition]{Prime}
A positive integer $p>1$ is \emph{prime} if the only divisors of $p$ are $1$ and $p$.
\domino{$p>1$ is \emph{prime}}{$\iff$}{$a$ has exactly two positive divisors: 1 and $p$}
\end{flashcard}

\begin{flashcard}[Definition]{Composite}
A positive integer $a$ is \emph{composite} if there exists a positive integer $b>1$ satisfying $b\mid a$.
\domino{$a>1$ is composite}{$\iff$}{$b\mid a$ and $1 < b < a$}
\end{flashcard}

\begin{flashcard}[Definition]{Greatest common divisor}
The \emph{greatest common divisor} of two integers $a$ and $b$, denoted $\gcd(a,b)$, is the largest integer that divides both $a$ and $b$.
\end{flashcard}

\begin{flashcard}[Definition]{Least common multiple}
The \emph{least common multiple} of two integers $a$ and $b$, denoted $\lcm(a,b)$, is the smallest positive integer is a multiple of both $a$ and $b$.
\end{flashcard}

\begin{flashcard}[Theorem]{Well-ordering principle}
Every non-empty subset of $\bbn$ contains a least element.
\end{flashcard}

\begin{flashcard}[Theorem]{Division algorithm}
Given integers $a$ and $b$ with $b>0$, there exist unique integers $q$ and $r$ that satisfy $a = bq+r$, where $0 \le r < b$.
\end{flashcard}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Graph theory
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{flashcard}[Definition]{Graph}
A \emph{graph} is a pair $G = (V,E)$, where $V$ is a nonempty finite set and $E$ is a set of two-element subsets of $V$. The elements in $V$ are \emph{vertices} and the elements of $E$ are \emph{edges}.
\end{flashcard}

\begin{flashcard}[Definition]{Adjacent}
Let $G=(V,E)$ be a graph. If $u,v \in V$, then $u$ is \emph{adjacent} to $v$ if $\{u,v\} \in E$. We also use the notation $u \sim v$ to denote that $u$ is adjacent to $v$.

\domino{$u \sim v$}{$\iff$}{$\{u,v\} \in E$}
\end{flashcard}

\begin{flashcard}[Definition]{Endpoint of an edge}
Let $G = (V,E)$ is a graph. If $\{u,v\} \in E$, then the \emph{endpoints} of $\{u,v\}$ are the vertices $u$ and $v$.
\end{flashcard}

\begin{flashcard}[Definition]{Incident}
Let $G = (V,E)$ be a graph. An vertex $v\in V$ is \emph{incident} with the edge $e\in E$ if $v \in e$.
\end{flashcard}

\begin{flashcard}[Definition]{Neighbors of a vertex}
Let $G = (V,E)$. For any $v \in V$, the \emph{neighbors} of $v$ is the set of vertices adjacent to $v$. The set of neighbors of $v$ is denoted by
\begin{align*}
N(v) = \{ u \in V : u \sim v\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Degree of a vertex}
Let $G = (V,E)$ be a graph. The \emph{degree} of a vertex $v \in V$ is the number of edges in $G$ incident with $v$, and is denoted by
\begin{align*}
d(v) &= \#\{e \in E : v \in e\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Theorem]{Graph relation}
For any graph $G = (V,E)$, we have 
\begin{align*}
\sum_{v \in V} d(v) = 2|E|.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Order of a graph}
The \emph{order} of a graph $G$ is $|V(G)|$, the number vertices in $G$.
\end{flashcard}

\begin{flashcard}[Definition]{Size of a graph}
The \emph{size} of a graph $G$ is $|E(G)|$, the number edges in $G$.
\end{flashcard}

\begin{flashcard}[Definition]{Maximum and minimum degree}
Let $G = (V,E)$ be a graph. The \emph{maximum degree} and \emph{minimum degree} of the vertices in $G$ are
\begin{align*}
\Delta(G) &= \max \{d(v) : v \in V\}\\
\delta(G) &= \min \{d(v) : v \in V\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Regular / $k$-regular graph}
A graph $G=(V,E)$ is \emph{regular} if all of the vertices in the graph has the same degree. Moreover, the graph is \emph{$k$-regular} if $d(v) = k$ for all $v \in V$.

\domino{$G$ is $k$-regular}{$\iff$}{$d(v) = k$ for all $v \in V(G)$}
\end{flashcard}

\begin{flashcard}[Definition]{Complete graph}
A graph is \emph{complete} if each vertex is adjacent to each other vertex. The complete graph of order $n$ is denoted by $K_n$. 
\end{flashcard}

\begin{flashcard}[Definition]{Subgraph}
Let $G$ and $H$ be graphs. The graph $H$ is a \emph{subgraph} of $G$ if $V(H) \subseteq V(G)$ and $E(H) \subseteq E(G)$.
\end{flashcard}

\begin{flashcard}[Definition]{Edge deletion}
Let $G$ be a graph. An \emph{edge deletion} is the process of removing an edge $e$ from $G$. This process results in a new graph, denoted by $G-e$, where
\begin{align*} 
V(G-e) = V(G) \atext{and} E(G-e) = E(G)-\{e\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Spanning subgraph}
Let $G$ and $H$ be graphs. The graph $H$ is a \emph{spanning subgraph} of $G$ if $H$ is a subgraph of $G$ and $V(H) = V(G)$.
\end{flashcard}

\begin{flashcard}[Definition]{Vertex deletion}
Let $G$ be a graph. A \emph{vertex deletion} is the process of removing a vertex $v$ from $G$ to create a new graph, denoted by $G-v$, where
\begin{align*}
V(G-v) &= V(G) - \{v\}\\
E(G-v) &= \{e \in E(G) : v \not\in e\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Induced subgraph}
Let $G$ be a graph and let $W$ be a subset of vertices is $G$ (that is, $W \subseteq V(G)$). The \emph{induced subgraph} of $G$ onto $W$ is the new graph, denoted by $G[W]$, where
\begin{align*}
V(G[W]) &= W,\\
E(G[W]) &= \{\{u,v\} \in E(G) : u \in W, v \in W\}.
\end{align*}
\end{flashcard}

\begin{flashcard}[Definition]{Clique / clique number}
Let $G$ be a graph. A subset of vertices $W \subseteq V(G)$ is called a \emph{clique} if $G[W]$ is a complete graph. The \emph{clique number} of $G$ is the size of the largest clique in $G$.
\end{flashcard}

\begin{flashcard}[Definition]{Independent set / independence number}
Let $G$ be a graph. A subset of vertices $W \subseteq V(G)$ is \emph{independent} if no two vertices in $W$ are adjacent. The \emph{independence number} of $G$ is the size of the largest independent set in $G$.
\end{flashcard}

\begin{flashcard}[Definition]{Walk / $(u,v)$-walk}
Let $G$ be a graph. A \emph{walk} in $G$ is a sequence of vertices $W = (v_0, v_1, v_2, \dotsc, v_n)$ where each vertex adjacent to the next, that is, $v_0 \sim v_1 \sim v_2 \sim \dotsb \sim v_n$.

A \emph{$(u,v)$-walk} is a walk whose first vertex is $u$ and whose last vertex is $v$.
\end{flashcard}

\begin{flashcard}[Definition]{Walk length}
The \emph{length} of a walk is the number of edges traversed by the walk. 
\end{flashcard}

\begin{flashcard}[Definition]{Concatenation}
Let $W_1$ be a $(u,v)$-walk and $W_2$ be a $(v,w)$-walk. The \emph{concatenation} of $W_1$ and $W_2$ (denoted by $W_1 + W_2$) is the $(u,w)$-walk defined by the walk $W_1$, followed by the walk $W_2$.
\end{flashcard}

\begin{flashcard}[Definition]{Path / $(u,v)$-path}
A \emph{path} in a graph is a walk in which no vertex is repeated. A $(u,v)$-path is a path from $u$ to $v$.
\end{flashcard}

\begin{flashcard}[Definition]{Connected vertices}
Let $G$ be a graph. The vertices $u, v \in V(G)$ are \emph{connected} if $G$ contains a $(u,v)$-path.
\domino{$u$ is connected to $v$}{$\iff$}{$G$ contains a $(u,v)$-path}
\end{flashcard}

\begin{flashcard}[Definition]{Connected graph}
A graph $G$ is \emph{connected} if, for every pair of vertices $u,v \in V(G)$, $u$ and $v$ are connected.
\domino{$G$ is connected}{$\iff$}{for every $u,v \in V(G)$, $u$ is connected to $v$}
\end{flashcard}

\begin{flashcard}[Definition]{Component}
A \emph{component} $H$ of a graph $G$ is a maximal connected subgraph of $G$, meaning that $H$ is not a proper subgraph of any connected subgraph of $G$.
\end{flashcard}

\begin{flashcard}[Definition]{Cut vertex}
Let $G$ be a graph. A vertex $v \in V(G)$ is a \emph{cut vertex} if $G-v$ has more components than $G$.
\domino{$v$ is a cut vertex}{$\iff$}{$G-v$ has more components than $G$}
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\begin{flashcard}[Definition]{Cut edge}
Let $G$ be a graph. An edge $e \in E(G)$ is a \emph{cut edge} if $G-e$ has more components than $G$.
\domino{$v$ is a cut edge}{$\iff$}{$G-e$ has more components than $G$}
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\begin{flashcard}[Definition]{Cycle}
A \emph{cycle} is a walk of length at least three whose first vertex and last vertex are the same.
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\begin{flashcard}[Definition]{Tree}
A connected graph $G$ is a \emph{tree} if it does not contain any cycles.
\domino{$G$ is a tree}{$\iff$}{$G$ does not contain any cycles}
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\begin{flashcard}[Definition]{Forest}
A graph $G$ is a \emph{forest} if every component of $G$ is a tree.
\domino{$G$ is a forest}{$\iff$}{each component of $G$ is a tree}
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\begin{flashcard}[Definition]{Spanning tree}
Let $G$ be a graph. A \emph{spanning tree} of $G$ is a spanning subgraph of $G$ that is a tree.
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\begin{flashcard}[Definition]{Hamiltonian cycle / Hamiltonian graph}
Let $G$ be a graph. A \emph{Hamiltonian cycle} is a cycle of $G$ that contains all of the vertices of $G$. A graph $G$ is \emph{Hamiltonian} if it contains a Hamiltonian cycle.
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\begin{flashcard}[Definition]{Eulerian trail / Eulerian graph}
Let $G$ be a graph. An \emph{Eulerian trail} is a walk in $G$ that traverses every edge exactly once. A graph $G$ is \emph{Eulerian} if it contains an Eulerian trail.
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