SetAttributes[set, {Flat, Orderless}]

set[args___] := Block[{union},
  union = Union[{args}];
  (
    set @@ union
  ) /; Length[union] != Length[{args}]
]

set /: Union[s___set] := set[s]
set /: Element[e_, s_set] := MemberQ[s, e]
set /: Subset[s___set] := VectorQ[Partition[Reverse[{s}], 2, 1], SubsetQ @@ # &]
cardinality[s_set] := Length[s]
cardinality[_] = 0;

set /: MakeBoxes[s : set[args__], fmt_] :=
  MakeBoxes[Interpretation[〈args〉, s], fmt]

set /: MakeBoxes[set[], fmt_] := MakeBoxes[Interpretation["∅", set[]], fmt]

set[1, 1, 2, 2, 1, 3]
(* 〈1, 2, 3〉 *)

set[1, 2, 3] ⋃ set[3, 4, 5] ⋃ set[4, 5, 6]
(* 〈1, 2, 3, 4, 5, 6〉 *)

set[1, 2, 3] ⋂ set[3, 4, 5]
(* 〈3〉 *)

set[1, 2, 3] ⋂ set[4, 5, 6]
(* ∅ *)

set[1, 2, 3] ⋂ set[3, 4, 5] ⋃ set[4, 8]
(* 〈3, 4, 8〉 *)

5 ∈ set[1, 2, 3, 4, 5]
(* True *)

set[1, 2, 3] ⊂ set[1, 2, 3, 4] ⊂ set[1, 2, 3, 4, 5]
(* True *)

cardinality[set[1, 2, 3, 4, 5]]
(* 5 *)

MakeBoxes[EmptySet, StandardForm] ^= TemplateBox[
    {},
    "EmptySet",
    DisplayFunction -> ("[EmptySet]"&),
    InterpretationFunction -> ("EmptySet"&)
];
MakeBoxes[UniversalSet, StandardForm] ^= TemplateBox[
    {},
    "UniversalSet",
    DisplayFunction -> (StyleBox["[DoubleStruckCapitalU]", FontFamily->"Times"]&),
    InterpretationFunction -> ("UniversalSet"&)
];

CurrentValue[EvaluationNotebook[], InputAliases] = {
    "su" -> "⊔",
    "si" -> "⊓",
    "sb" -> "⊏",
    "es" -> TemplateBox[
        {},
        "EmptySet",
        DisplayFunction -> ("[EmptySet]"&),
        InterpretationFunction -> ("EmptySet"&)
    ],
    "us" -> TemplateBox[
        {},
        "UniversalSet",
        DisplayFunction -> (StyleBox["[DoubleStruckCapitalU]", FontFamily->"Times"]&),
        InterpretationFunction -> ("UniversalSet"&)
    ]
};

toBoolean[expr_] := ReplaceAll[
    expr,
    {
        SquareUnion -> Or,
        SquareIntersection -> And,
        SquareSubset -> Implies,
        Backslash -> (And[#1, !#2]&),
        OverBar -> Not,
        Equal -> Equivalent,
        EmptySet -> False,
        UniversalSet -> True
    }
]

set = A ⊓ B ⊔ B;

toBoolean[set]

BooleanMinimize[toBoolean[set]]

set = A ⊓ B;
cond = A ⊏ B;

BooleanMinimize[toBoolean[set], toBoolean[cond]]

setQ[EmptySet] = True;
setQ[UniversalSet] = True;
setQ[_Symbol] = True;

setQ[(Backslash | SquareUnion | SquareIntersection | OverBar)[a__]] := AllTrue[{a}, setQ]

setQ[_] = False;

setQ[A ⊔ B]
setQ[A ⊏ B]
setQ[A == B]

fromBoolean[expr_] := ReplaceAll[
    expr,
    {
    Or -> SquareUnion,
    And -> SquareIntersection,
    Not -> OverBar,
    Equivalent -> Equal
    }
]
fromBoolean[a_ && !b_Symbol] := Backslash[a, b]
fromBoolean[!a_Symbol && b_] := Backslash[b, a]
fromBoolean[a_ || !b_Symbol] := SquareSubset[b, a]
fromBoolean[!a_Symbol || b_] := SquareSubset[a, b]

Options[SetSimplify] = {Method -> Automatic};

SetSimplify[set_, cond_:True, OptionsPattern[]] := Module[{res},
    res = fromBoolean @ BooleanMinimize[
        toBoolean[set],
        toBoolean[cond],
        Method->OptionValue[Method]
    ];
    If[setQ[set],
        res /. {False -> EmptySet, True -> UniversalSet},
        res
    ]
]

SetSimplify[(A ⊓ B) ⊔ B]
SetSimplify[A ⊓ B == A, A ⊏ B]
SetSimplify[A ⊔ B == B, A ⊏ B]

set = A ⊓ UniversalSet;
set //TeXForm

SetSimplify[set]

set = A ⊔ OverBar[A];
set //TeXForm

SetSimplify[set] //TeXForm

SetSimplify[A ⊔ A]
SetSimplify[A ⊓ A]

SetSimplify[A ⊔ (A ⊓ B)]
SetSimplify[A ⊓ (A ⊔ B)]

law = OverBar[A ⊔ B] == OverBar[A] ⊓ OverBar[B];
law //TeXForm

SetSimplify[law]

set = OverBar[EmptySet];
set //TeXForm

SetSimplify[set] //TeXForm

SetSimplify[A ⊏ A]

SetSimplify[A == B, A ⊏ B && B ⊏ A]

SetSimplify[A ⊏ C, A ⊏ B && B ⊏ C]

SetSimplify[A ⊏ A ⊔ B]
SetSimplify[A ⊔ B ⊏ C, A ⊏ C && B ⊏ C]

SetSimplify[A ⊓ B ⊏ A]
SetSimplify[C ⊏ A ⊓ B, C ⊏ A && C ⊏ B]

SetSimplify[A ⊓ B == A, A ⊔ B == B]

SetSimplify[A ∖ B, A ⊏ B] //TeXForm

SetSimplify[A ∖ A] //TeXForm

set = UniversalSet ∖ A;
set //TeXForm

SetSimplify[set] //TeXForm

MakeBoxes[EmptySet, form_] ^= TemplateBox[
    {},
    "EmptySet",
    DisplayFunction -> ("[EmptySet]"&),
    InterpretationFunction -> ("EmptySet"&)
];
MakeBoxes[UniversalSet, form_] ^= TemplateBox[
    {},
    "UniversalSet",
    DisplayFunction -> (StyleBox["[DoubleStruckCapitalU]", FontFamily->"Times"]&),
    InterpretationFunction -> ("UniversalSet"&)
];

CurrentValue[EvaluationNotebook[], InputAliases] = {
    "su" -> "⊔",
    "si" -> "⊓",
    "sb" -> "⊏",
    "es" -> TemplateBox[
        {},
        "EmptySet",
        DisplayFunction -> ("[EmptySet]"&),
        InterpretationFunction -> ("EmptySet"&)
    ],
    "us" -> TemplateBox[
        {},
        "UniversalSet",
        DisplayFunction -> (StyleBox["[DoubleStruckCapitalU]", FontFamily->"Times"]&),
        InterpretationFunction -> ("UniversalSet"&)
    ]
};

toBoolean[expr_] := ReplaceAll[
    expr,
    {
        SquareUnion -> Or,
        SquareIntersection -> And,
        SquareSubset -> Implies,
        OverBar -> Not,
        Backslash -> (And[#1, !#2]&),
        Equal -> Equivalent,
        EmptySet -> False,
        UniversalSet -> True
    }
]

setQ[EmptySet] = True;
setQ[UniversalSet] = True;
setQ[_Symbol] = True;

setQ[(Backslash | SquareUnion | SquareIntersection | OverBar)[a__]] := AllTrue[{a}, setQ]

setQ[_] = False;

fromBoolean[expr_] := ReplaceAll[
    expr,
    {
    Or -> SquareUnion,
    And -> SquareIntersection,
    Not -> OverBar,
    Equivalent -> Equal
    }
]
fromBoolean[a_ && !b_Symbol] := Backslash[a, b]
fromBoolean[!a_Symbol && b_] := Backslash[b, a]
fromBoolean[a_ || !b_Symbol] := SquareSubset[b, a]
fromBoolean[!a_Symbol || b_] := SquareSubset[a, b]

Options[SetSimplify] = {Method -> Automatic};

SetSimplify[set_, cond_:True, OptionsPattern[]] := Module[{res},
    res = fromBoolean @ BooleanMinimize[
        toBoolean[set],
        toBoolean[cond],
        Method->OptionValue[Method]
    ];
    If[setQ[set],
        res /. {False -> EmptySet, True -> UniversalSet},
        res
    ]
]