A sentence that is either true or false, but not both.

A proposition that is always false.

A proposition that is neither a tautology nor a contradiction.

A proposition that is always true.

Every even integer greater than 2 can be expressed as the sum of two prime numbers.

A table used to determine the truth value of a logical expression based on all possible combinations of input truth values.

$p$ AND $q$

$p$ OR $q$

NOT

If $p$, then $q$ | $p$ implies $q$

$p$ if and and only if $q$

$p$ and $q$ have identical truth values.

$\begin{array}{l}p\vee q\equiv q\vee p\\ p\wedge q\equiv q\wedge p\end{array}$

$\begin{array}{l}(p\vee q)\vee r\equiv p\vee (q\vee r)\\ (p\wedge q)\wedge r\equiv p\wedge (q\wedge r)\end{array}$

$\begin{array}{l}p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)\\ p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)\end{array}$

$\begin{array}{l}p\vee F\equiv p\\ p\wedge T\equiv p\end{array}$

$\begin{array}{l}p\wedge \mathrm{\neg }p\equiv F\\ p\vee \mathrm{\neg }p\equiv T\end{array}$

$\begin{array}{l}p\vee p\equiv p\\ p\wedge p\equiv p\end{array}$

$\begin{array}{l}p\wedge F\equiv F\\ p\vee T\equiv T\end{array}$

$\begin{array}{l}p\wedge (p\vee q)\equiv p\\ p\vee (p\wedge q)\equiv p\end{array}$

$\begin{array}{l}\mathrm{\neg }(p\vee q)\equiv (\mathrm{\neg }p)\wedge (\mathrm{\neg }q)\\ \mathrm{\neg }(p\wedge q)\equiv (\mathrm{\neg }p)\vee (\mathrm{\neg }q)\end{array}$

$\mathrm{\neg }(\mathrm{\neg }p)\equiv p$

$p\to q\equiv \mathrm{\neg }p\vee q$

$\begin{array}{l}p\to q\\ p\\ \therefore q\end{array}$

$\begin{array}{l}p\to q\\ \mathrm{\neg }q\\ \therefore \mathrm{\neg }p\end{array}$

$\begin{array}{l}p\vee q\\ \sim p\\ \therefore q\end{array}$

$\begin{array}{l}p\to q\\ q\to r\\ \therefore p\to r\end{array}$

$\begin{array}{l}p\to q\\ q\\ \therefore p\end{array}$

$\begin{array}{l}p\to q\\ \sim p\\ \therefore -q\end{array}$

For all $x$, $p(x)$

There exists $x$ such that $p(x)$

A collection of objects, enclosed in braces.

An object in a set.

The total number of elements in a set.

A special notation used to describe sets that are too complex to list between braces.

A set with a countable amount of elements.

A set with an infinite amount of elements.

A list $x,y$ of two things $x$ and $y$.

A set that has no elements.

The universal set.

Let $A$ be a set with a universal set $U$. The complement of $A$ is the set $\bar{A}=\mathcal{U}-A$.

The set of natural numbers (Positive Integers)

The set of integers

The set of rational numbers

The set of real numbers

PowerSet

Is an element of

Is an subset of

Is a proper subset of

Set Intersection

Set Union

The multiplication of two sets $A$ and $B$, resulting in a set of ordered pairs of elements from $A$ and $B$.

The cartesian product of a set $A$ with itself $n$ times.

The difference of two sets $A$ and $B$ is the set of all elements that are in $A$ but not in $B$.

An ordered sequence of objects, enclosed in parenthesis.

A special list with no entries.

Suppose in making a list of length $n$ there are $a_1$ possible choices for the first entry, $a_2$ possible choices for the second entry, $a_3$ possible choices for the third entry, and so on. Then the total number of different lists that can be made this way is the product of $a_1\cdot a_2\cdot a_3\cdots a_n$.