Set theory: The subset relation

The relation tells, that the set is empty:    =

In written formulas:

The relation ${\displaystyle A\subseteq B}$ tells, that the set ${\displaystyle A\cap B^{c}}$ is empty:   ${\displaystyle A\cap B^{c}=\emptyset }$

Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define the subset relation:

Venn diagrams	written formulas
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle A\setminus B=\emptyset }$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle A=A\cap B}$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle A\Delta B=B\setminus A}$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle A\cup B=B}$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle B^{c}=A^{c}\cap B^{c}}$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle A\cup B^{c}=(A\Delta B)^{c}}$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle A^{c}\cup B^{c}=A^{c}}$
${\displaystyle \Leftrightarrow }$    =	${\displaystyle A\subseteq B}$   ${\displaystyle \Leftrightarrow }$   ${\displaystyle \emptyset ^{c}=A^{c}\cup B}$

The sign ${\displaystyle \Leftrightarrow }$ tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.

Propositional logic: The logical implication

The relation tells, that the statement is never true:   ${\displaystyle \Leftrightarrow }$

In written formulas:

The relation ${\displaystyle A\Rightarrow B}$ tells, that the statement ${\displaystyle A\land \neg B}$ is never true:   ${\displaystyle A\land \neg B\Leftrightarrow false}$

Under this condition, several logic operations, not equivalent in general, produce equivalent results.
These equivalences define the logical implication:

Venn diagrams	written formulas
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle A\land \neg B\Leftrightarrow false}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle A\Leftrightarrow A\land B}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle A\oplus B\Leftrightarrow \neg A\land B}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle A\lor B\Leftrightarrow B}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle \neg B\Leftrightarrow \neg A\land \neg B}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle A\leftarrow B\Leftrightarrow A\leftrightarrow B}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle \neg A\lor \neg B\Leftrightarrow \neg A}$
${\displaystyle \equiv }$   ${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow B}$   ${\displaystyle \equiv }$   ${\displaystyle true\Leftrightarrow A\rightarrow B}$

Especially the last line in this table is important:
The logical implication ${\displaystyle A\Rightarrow B}$ tells, that the material implication ${\displaystyle A\rightarrow B}$ is always true.
The material implication ${\displaystyle A\rightarrow B}$ is the same as ${\displaystyle \neg A\lor B}$.
Note: Names like logical implication and material implication are used in many different ways, and shouldn't be taken too serious.

The sign ${\displaystyle \equiv }$ tells, that two statements about statements about whatever objects mean the same.
The sign ${\displaystyle \Leftrightarrow }$ tells, that two statements about whatever objects mean the same.

Important relations

Set theory:	subset	disjoint	subdisjoint	equal	complementary
Logic:	implication	contrary	subcontrary	equivalent	contradictory