The **implication** sign requires that if $p$ is true, $q$ is true, such that *$p$ implies $q$*. The first symbol is the **hypothesis** and the second symbol is the **conclusion**.
The **inference** sign represents the inverse of the implication sign, such that $p$ **is implied by** $q$. It is equivalent to $q\implies p$.
$$p\impliedby q$$
The **if and only if** sign requires that the two propositions imply each other — i.e., that the state of $p$ is the same as the state of $q$. It is equivalent to $(p\implies q)\wedge (p\impliedby q)$.
$$p\iff q$$
The **logical equivalence** sign represents if the truth values for both statements are **the same for all possible variables**, such that the two are **equivalent statements**.
$$p\equiv q$$
$p\equiv q$ can also be defined as true when $p\iff q$ is a tautology.
!!! warning
$p\equiv q$ is *not a proposition* itself but instead *describes* propositions. $p\iff q$ is the propositional equivalent.
## Common theorems
The **double negation rule** states that if $p$ is a proposition:
$$\neg(\neg p)\equiv p$$
!!! tip "Proof"
Note that:
| $p$ | $\neg p$ | $\neg(\neg p)$
| --- | --- | --- |
| T | F | T |
| F | T | F |
Because the truth values of $p$ and $\neg(\neg p)$ for all possible truth values are equal, by definition, it follows that $p\equiv\neg(\neg p)$.
!!! warning
Proofs must include the definition of what is being proven, and any relevant evidence must be used to describe why.
The two **De Morgan's Laws** allow distributing the negation operator in a dis/conjunction if the junction is inverted.
$$
\neg(p\vee q)\equiv(\neg p)\wedge(\neg q) \\
\neg(p\wedge q)\equiv(\neg p)\vee(\neg q)
$$
An implication can be expressed as a disjunction. As long as it is stated, it can used as its definition.
