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The rounding and congruence functions  ,  ,  ,  ,  ,  , and  have numerous representations through related functions, which are shown in the following tables, where the symbol  means the characteristic function of a set  (having the value 1 when its argument  is an element of the specified set 𝔸, and a value of 0 otherwise):
The rounding and congruence functions  ,  ,  ,  ,  , and  can also be represented through elementary functions by the following formulas:
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