Set theory: The complementary relation

Two sets ${\displaystyle ~A}$ and ${\displaystyle ~B}$ are complementary,
when they are disjoint and subdisjoint (no elements are inside both and no elements are outside both of them),
so when all elements are either in set ${\displaystyle ~A}$ or in set ${\displaystyle ~B}$.
In other words: When the complement of their symmetric difference is empty.

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

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

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

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

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

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

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