Set theory: The equivalence of sets

Two sets ${\displaystyle ~A}$ and ${\displaystyle ~B}$ are equivalent - i.e. contain the same elements - when all elements of ${\displaystyle ~A}$ are in ${\displaystyle ~B}$, and all elements of ${\displaystyle ~B}$ are in ${\displaystyle ~A}$.
In other words: If their symmetric difference is empty.

	${\displaystyle \Leftrightarrow }$		${\displaystyle \land }$		${\displaystyle \Leftrightarrow }$		=
${\displaystyle ~A=B~}$	${\displaystyle \Leftrightarrow }$	${\displaystyle A\subseteq B}$	${\displaystyle \land }$	${\displaystyle B\subseteq A}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\Delta B}$	=	${\displaystyle ~\emptyset }$

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

	${\displaystyle \Leftrightarrow }$		=		=		=
${\displaystyle ~A=B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\Delta B}$	=	${\displaystyle ~A\setminus B}$	=	${\displaystyle ~B\setminus A}$	=	${\displaystyle \emptyset }$

	${\displaystyle \Leftrightarrow }$		=		=		=
${\displaystyle ~A=B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\cup B}$	=	${\displaystyle ~A}$	=	${\displaystyle ~B}$	=	${\displaystyle ~A\cap B}$

	${\displaystyle \Leftrightarrow }$		=		=		=
${\displaystyle ~A=B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A^{c}\cup B^{c}}$	=	${\displaystyle ~B^{c}}$	=	${\displaystyle ~A^{c}}$	=	${\displaystyle ~A^{c}\cap B^{c}}$

	${\displaystyle \Leftrightarrow }$		=		=		=
${\displaystyle ~A=B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~\emptyset ^{c}}$	=	${\displaystyle ~A\cup B^{c}}$	=	${\displaystyle ~A^{c}\cup B}$	=	${\displaystyle ~(A\Delta B)^{c}}$

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 equivalence of statements

Two statements ${\displaystyle ~A}$ and ${\displaystyle ~B}$ are equivalent - i.e. together true or together false - when ${\displaystyle ~A}$ implies ${\displaystyle ~B}$, and ${\displaystyle ~B}$ implies ${\displaystyle ~A}$.
In other words: If their exclusive or is never true.

	${\displaystyle \equiv }$		${\displaystyle \land }$		${\displaystyle \equiv }$		${\displaystyle \Leftrightarrow }$
${\displaystyle ~A\Leftrightarrow B~}$	${\displaystyle \equiv }$	${\displaystyle A\Rightarrow B}$	${\displaystyle \land }$	${\displaystyle B\Rightarrow A}$	${\displaystyle \equiv }$	${\displaystyle ~A\oplus B}$	${\displaystyle ~\Leftrightarrow }$	${\displaystyle ~false}$

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

	${\displaystyle \equiv }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$
${\displaystyle ~A\Leftrightarrow B}$	${\displaystyle \equiv }$	${\displaystyle ~A\oplus B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\land \neg B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle \neg A\land B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~false}$

	${\displaystyle \equiv }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$
${\displaystyle ~A\Leftrightarrow B}$	${\displaystyle \equiv }$	${\displaystyle ~A\lor B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\land B}$

	${\displaystyle \equiv }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$
${\displaystyle ~A\Leftrightarrow B}$	${\displaystyle \equiv }$	${\displaystyle ~\neg A\lor \neg B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~\neg B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~\neg A}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~\neg A\land \neg B}$

	${\displaystyle \equiv }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$
${\displaystyle ~A\Leftrightarrow B}$	${\displaystyle \equiv }$	${\displaystyle ~true}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\lor \neg B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle \neg A\lor B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~A\leftrightarrow B}$

Especially the last line is important:
The logical equivalence ${\displaystyle A\Leftrightarrow B}$ tells, that the material equivalence ${\displaystyle A\leftrightarrow B}$ is always true.
The material equivalence ${\displaystyle A\leftrightarrow B}$ is the same as ${\displaystyle \neg (A\oplus B)}$, the negated exclusive or.
Note: Names like logical equivalence and material equivalence 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