#
# Mathematical operators
# Range: 2200–22FF
#

# Miscellaneous mathematical symbols
<Multi_key> <Multi_key> <A> <A>                                       : "∀"  U2200    # FOR ALL
                                                                                      # = universal quantifier
<Multi_key> <Multi_key> <p> <a> <r> <t>                               : "∂"  U2202    # PARTIAL DIFFERENTIAL
<Multi_key> <Multi_key> <p> <d> <i> <f>                               : "∂"  U2202    # PARTIAL DIFFERENTIAL
<Multi_key> <Multi_key> <E> <E>                                       : "∃"  U2203    # THERE EXISTS
                                                                                      # = existential quantifier
<Multi_key> <Multi_key> <slash> <E> <E>                               : "∄"  U2204    # THERE DOES NOT EXIST
<Multi_key> <Multi_key> <l> <a> <p>                                   : "∆"  U2206    # INCREMENT
                                                                                      # = Laplace operator
                                                                                      # = forward difference
                                                                                      # = symmetric difference (in set theory)
<Multi_key> <Multi_key> <d> <i> <f>                                   : "∆"  U2206    # INCREMENT
<Multi_key> <Multi_key> <d> <i> <f> <f>                               : "∆"  U2206    # INCREMENT
<Multi_key> <Multi_key> <d> <e> <l>                                   : "∇"  U2207    # NABLA
                                                                                      # = backward difference
                                                                                      # = gradient, del

# Set membership
<Multi_key> <Multi_key> <i> <n>                                       : "∈"  U2208    # ELEMENT OF
<Multi_key> <Multi_key> <exclam> <i> <n>                              : "∉"  U2209    # NOT AN ELEMENT OF
<Multi_key> <Multi_key> <slash> <i> <n>                               : "∉"  U2209    # NOT AN ELEMENT OF
<Multi_key> <Multi_key> <n> <i>                                       : "∋"  U220B    # CONTAINS AS MEMBER
                                                                                      # = such that
<Multi_key> <Multi_key> <exclaim> <n> <i>                             : "∌"  U220C    # DOES NOT CONTAIN AS MEMBER
<Multi_key> <Multi_key> <slash> <n> <i>                               : "∌"  U220C    # DOES NOT CONTAIN AS MEMBER

# Miscellaneous mathematical symbol
<Multi_key> <Multi_key> <Q> <E> <D>                                   : "∎"  U220E    # END OF PROOF
                                                                                      # = q.e.d.
<Multi_key> <Multi_key> <q> <e> <d>                                   : "∎"  U220E    # END OF PROOF

# N-ary operators
<Multi_key> <Multi_key> <p> <r> <o> <d>                               : "∏"  U220F    # N-ARY PRODUCT
                                                                                      # = product sign
<Multi_key> <Multi_key> <s> <u> <m>                                   : "∑"  U2211    # N-ARY SUMMATION
                                                                                      # = summation sign

# Operators
<Multi_key> <Multi_key> <minus> <period>                              : "−"  U2212    # MINUS
<Multi_key> <Multi_key> <minus> <plus>                                : "∓"  U2213    # MINUS-OR-PLUS SIGN
<Multi_key> <Multi_key> <slash> <slash>                               : "∕"  U2215    # DIVISION SLASH
<Multi_key> <Multi_key> <d> <i> <v>                                   : "∕"  U2215    # DIVISION SLASH
                                                                                      # • generic division operator
<Multi_key> <Multi_key> <asterisk> <o>                                : "∘"  U2218    # RING OPERATOR
                                                                                      # = composite function
                                                                                      # = APL jot
# SQUARE ROOT exists as:
#     <Multi_key> <v> <slash>
#     <Multi_key> <slash> <v>
<Multi_key> <Multi_key> <3> <v> <slash>                               : "∛"  U221B    # CUBE ROOT
<Multi_key> <Multi_key> <4> <v> <slash>                               : "∜"  U221C    # FOURTH ROOT
<Multi_key> <Multi_key> <8> <3>                                       : "∝"  U221D    # PROPORTIONAL TO

# Angles
<Multi_key> <Multi_key> <bar> <underscore>                            : "∟"  U221F    # RIGHT ANGLE
<Multi_key> <Multi_key> <slash> <underscore>                          : "∠"  U2220    # ANGLE
<Multi_key> <Multi_key> <parenright> <slash> <underscore>             : "∡"  U2221    # MEASURED ANGLE

# Logical and set operators
<Multi_key> <Multi_key> <slash> <backslash>                           : "∧"  U2227    # LOGICAL AND
                                                                                      # = wedge, conjunction
<Multi_key> <Multi_key> <backslash> <slash>                           : "∨"  U2228    # LOGICAL OR
                                                                                      # = vee, disjunction
<Multi_key> <Multi_key> <braceleft> <asciicircum>                     : "∩"  U2229    # INTERSECTION
                                                                                      # = cap, hat
<Multi_key> <Multi_key> <braceleft> <U>                               : "∪"  U222A    # UNION
                                                                                      # = cup

# Integrals
<Multi_key> <Multi_key> <i> <n> <t>                                   : "∫"  U222B    # INTEGRAL
<Multi_key> <Multi_key> <i> <i> <n> <t>                               : "∬"  U222C    # DOUBLE INTEGRAL
<Multi_key> <Multi_key> <i> <i> <i> <n> <t>                           : "∭"  U222D    # TRIPLE INTEGRAL
<Multi_key> <Multi_key> <o> <i> <n> <t>                               : "∮"  U222E    # CONTOUR INTEGRAL
<Multi_key> <Multi_key> <o> <i> <i> <n> <t>                           : "∯"  U222F    # SURFACE INTEGRAL
<Multi_key> <Multi_key> <o> <i> <i> <i> <n> <t>                       : "∰"  U2230    # VOLUME INTEGRAL

# Relations
<Multi_key> <Multi_key> <tilde> <period>                              : "∼"  U223C    # TILDE OPERATOR
                                                                                      # = varies with (proportional to)
                                                                                      # = difference between
                                                                                      # = similar to
                                                                                      # = not
                                                                                      # = cycle
                                                                                      # = APL tilde

# Miscellaneous mathematical symbol
<Multi_key> <Multi_key> <s> <i> <n> <e>                               : "∿"  U223F    # SINE WAVE
                                                                                      # = alternating current

# Relations
<Multi_key> <Multi_key> <slash> <tilde> <period>                      : "≁"  U2241    # NOT TILDE
<Multi_key> <Multi_key> <tilde> <slash>                               : "≁"  U2241    # NOT TILDE
<Multi_key> <Multi_key> <tilde> <minus>                               : "≃"  U2243    # ASYMPTOTICALLY EQUAL TO
<Multi_key> <Multi_key> <minus> <tilde>                               : "≃"  U2243    # ASYMPTOTICALLY EQUAL TO
<Multi_key> <Multi_key> <slash> <tilde> <minus>                       : "≄"  U2244    # NOT ASYMPTOTICALLY EQUAL TO
<Multi_key> <Multi_key> <tilde> <minus> <slash>                       : "≄"  U2244    # NOT ASYMPTOTICALLY EQUAL TO
<Multi_key> <Multi_key> <tilde> <equal>                               : "≅"  U2245    # APPROXIMATELY EQUAL TO
<Multi_key> <Multi_key> <equal> <tilde>                               : "≅"  U2245    # APPROXIMATELY EQUAL TO
<Multi_key> <Multi_key> <tilde> <slash> <equal>                       : "≆"  U2246    # APPROXIMATELY BUT NOT ACTUALLY EQUAL TO
<Multi_key> <Multi_key> <equal> <slash> <tilde>                       : "≆"  U2246    # APPROXIMATELY BUT NOT ACTUALLY EQUAL TO
<Multi_key> <Multi_key> <tilde> <equal> <slash>                       : "≇"  U2247    # NEITHER APPROXIMATELY NOR ACTUALLY EQUAL TO
<Multi_key> <Multi_key> <slash> <tilde> <equal>                       : "≇"  U2247    # NEITHER APPROXIMATELY NOR ACTUALLY EQUAL TO
<Multi_key> <Multi_key> <tilde> <tilde>                               : "≈"  U2248    # ALMOST EQUAL TO
                                                                                      # = asymptotic to
<Multi_key> <Multi_key> <slash> <tilde> <tilde>                       : "≉"  U2249    # NOT ALMOST EQUAL TO
<Multi_key> <Multi_key> <tilde> <tilde> <slash>                       : "≉"  U2249    # NOT ALMOST EQUAL TO
<Multi_key> <Multi_key> <tilde> <tilde> <minus>                       : "≊"  U224A    # ALMOST EQUAL OR EQUAL TO
<Multi_key> <Multi_key> <equal> <d> <e> <f>                           : "≝"  U225D    # EQUAL TO BY DEFINITION
<Multi_key> <Multi_key> <d> <e> <f> <equal>                           : "≝"  U225D    # EQUAL TO BY DEFINITION
<Multi_key> <Multi_key> <equal> <question>                            : "≟"  U225F    # QUESTIONED EQUAL TO
<Multi_key> <Multi_key> <equal> <minus>                               : "≡"  U2261    # IDENTICAL TO
<Multi_key> <Multi_key> <minus> <equal>                               : "≡"  U2261    # IDENTICAL TO
<Multi_key> <Multi_key> <slash> <equal> <minus>                       : "≢"  U2262    # NOT IDENTICAL TO
<Multi_key> <Multi_key> <slash> <minus> <equal>                       : "≢"  U2262    # NOT IDENTICAL TO
<Multi_key> <Multi_key> <equal> <minus> <slash>                       : "≢"  U2262    # NOT IDENTICAL TO
<Multi_key> <Multi_key> <minus> <equal> <slash>                       : "≢"  U2262    # NOT IDENTICAL TO
# LESS-THAN OR EQUAL TO and GREATER-THAN OR EQUAL TO exist as:
#     <Multi_key> <less> <underscore>                 : "≤"           U2264   # < _ LESS-THAN OR EQUAL TO
#     <Multi_key> <underscore> <less>                 : "≤"           U2264   # _ < LESS-THAN OR EQUAL TO
#     <Multi_key> <greater> <underscore>              : "≥"           U2265   # > _ GREATER-THAN OR EQUAL TO
#     <Multi_key> <underscore> <greater>              : "≥"           U2265   # _ > GREATER-THAN OR EQUAL TO
<Multi_key> <Multi_key> <less> <equal>                                : "≦"  U2266    # LESS-THAN OVER EQUAL TO
<Multi_key> <Multi_key> <equal> <less>                                : "≦"  U2266    # LESS-THAN OVER EQUAL TO
<Multi_key> <Multi_key> <greater> <equal>                             : "≧"  U2267    # GREATER-THAN OVER EQUAL TO
<Multi_key> <Multi_key> <equal> <greater>                             : "≧"  U2267    # GREATER-THAN OVER EQUAL TO
<Multi_key> <Multi_key> <less> <slash> <equal>                        : "≨"  U2268    # LESS-THAN BUT NOT EQUAL TO
<Multi_key> <Multi_key> <equal> <slash> <less>                        : "≨"  U2268    # LESS-THAN BUT NOT EQUAL TO
<Multi_key> <Multi_key> <greater> <slash> <equal>                     : "≩"  U2269    # GREATER-THAN BUT NOT EQUAL TO
<Multi_key> <Multi_key> <equal> <slash> <greater>                     : "≩"  U2269    # GREATER-THAN BUT NOT EQUAL TO
<Multi_key> <Multi_key> <plus> <less>                                 : "≪"  U226A    # MUCH LESS-THAN
<Multi_key> <Multi_key> <plus> <greater>                              : "≫"  U226B    # MUCH GREATER-THAN
<Multi_key> <Multi_key> <slash> <less>                                : "≮"  U226E    # NOT LESS-THAN
<Multi_key> <Multi_key> <less> <slash> <period>                       : "≮"  U226E    # NOT LESS-THAN
<Multi_key> <Multi_key> <slash> <greater>                             : "≮"  U226E    # NOT GREATER-THAN
<Multi_key> <Multi_key> <greater> <slash> <period>                    : "≮"  U226E    # NOT GREATER-THAN
<Multi_key> <Multi_key> <slash> <less> <underscore>                   : "≰"  U2270    # NEITHER LESS-THAN NOR EQUAL TO
<Multi_key> <Multi_key> <underscore> <less> <slash>                   : "≰"  U2270    # NEITHER LESS-THAN NOR EQUAL TO
<Multi_key> <Multi_key> <slash> <greater> <underscore>                : "≱"  U2271    # NEITHER GREATER-THAN NOR EQUAL TO
<Multi_key> <Multi_key> <underscore> <greater> <slash>                : "≱"  U2271    # NEITHER GREATER-THAN NOR EQUAL TO
<Multi_key> <Multi_key> <tilde> <less>                                : "≲"  U2272    # LESS-THAN OR EQUIVALENT TO
<Multi_key> <Multi_key> <less> <tilde>                                : "≲"  U2272    # LESS-THAN OR EQUIVALENT TO
<Multi_key> <Multi_key> <tilde> <greater>                             : "≳"  U2273    # GREATER-THAN OR EQUIVALENT TO
<Multi_key> <Multi_key> <greater> <tilde>                             : "≳"  U2273    # GREATER-THAN OR EQUIVALENT TO
<Multi_key> <Multi_key> <braceleft> <parenleft>                       : "⊂"  U2282    # SUBSET OF
                                                                                      # = included in set
<Multi_key> <Multi_key> <braceleft> <parenright>                      : "⊃"  U2283    # SUPERSET OF
                                                                                      # = includes in set
<Multi_key> <Multi_key> <slash> <braceleft> <parenleft>               : "⊄"  U2284    # NOT A SUBSET OF
<Multi_key> <Multi_key> <slash> <braceleft> <parenright>              : "⊅"  U2285    # NOT A SUPERSET OF
<Multi_key> <Multi_key> <braceleft> <underscore> <parenleft>          : "⊆"  U2286    # SUBSET OF OR EQUAL TO
<Multi_key> <Multi_key> <braceleft> <underscore> <parenright>         : "⊇"  U2287    # SUPERSET OF OR EQUAL TO
<Multi_key> <Multi_key> <slash> <braceleft> <underscore> <parenleft>  : "⊈"  U2288    # NEITHER A SUBSET OF NOR EQUAL TO
<Multi_key> <Multi_key> <slash> <braceleft> <underscore> <parenright> : "⊉"  U2289    # NEITHER A SUPERSET OF NOR EQUAL TO
<Multi_key> <Multi_key> <braceleft> <underscore> <slash> <parenleft>  : "⊊"  U228A    # SUBSET OF WITH NOT EQUAL TO
<Multi_key> <Multi_key> <braceleft> <underscore> <slash> <parenright> : "⊋"  U228B    # SUPERSET OF WITH NOT EQUAL TO

# Operators
<Multi_key> <Multi_key> <bar> <minus>                                 : "⊢"  U22A2    # RIGHT TACK
                                                                                      # = turnstile
                                                                                      # = proves, implies, yields
                                                                                      # = reducible
<Multi_key> <Multi_key> <minus> <bar>                                 : "⊣"  U22A2    # LEFT TACK
                                                                                      # = reverse turnstile
                                                                                      # = non-theorem, does not yield

# Miscellaneous mathematical symbols
<Multi_key> <Multi_key> <minus> <bar> <minus>                         : "⊤"  U22A4    # DOWN TACK
                                                                                      # = top
<Multi_key> <Multi_key> <underscore> <bar> <underscore>               : "⊥"  U22A5    # UP TACK
                                                                                      # = base, bottom

# Operators
<Multi_key> <Multi_key> <v> <underscore>                              : "⊻"  U22BB    # XOR
<Multi_key> <Multi_key> <underscore> <v>                              : "⊻"  U22BB    # XOR
<Multi_key> <Multi_key> <backslash> <underscore> <slash>              : "⊻"  U22BB    # XOR
<Multi_key> <Multi_key> <asciicircum> <minus>                         : "⊼"  U22BC    # NAND
<Multi_key> <Multi_key> <minus> <asciicircum>                         : "⊼"  U22BC    # NAND
<Multi_key> <Multi_key> <slash> <minus> <backslash>                   : "⊼"  U22BC    # NAND

# Operators
<Multi_key> <Multi_key> <asterisk> <period>                           : "⋅"  U22C5    # DOT OPERATOR
                                                                                      # • preferred to 00B7  ·   for denotation of
                                                                                      # multiplication

# Relations
<Multi_key> <Multi_key> <bar> <period> <period>                       : "⋮"  U22EE    # VERTICAL ELLIPSIS
                                                                                      # • these four ellipses are used for matrix
                                                                                      # row/column elision
<Multi_key> <Multi_key> <minus> <period> <period>                     : "⋯"  U22EF    # MIDLINE HORIZONTAL ELLIPSIS
<Multi_key> <Multi_key> <slash> <period> <period>                     : "⋰"  U22F0    # UP RIGHT DIAGONAL ELLIPSIS
<Multi_key> <Multi_key> <backslash> <period> <period>                 : "⋱"  U22F1    # DOWN RIGHT DIAGONAL ELLIPSIS