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# basic

Basic notations View other versions (2)

## Basic

 This defines a function whose arguments take values from the set and which returns a unique value from the set for each corresponding argument. For example consider the function , given by . By evaluating this function at various values of the argument the following results are obtained, , , . This defines the composite function defined by the relation for all values of , where and are apropriately chosen functions. The absolute value function defined as: For example , , . This tells that has an approximate value of , in other words where is a very small positive value. For example . This tells that has a much greater value than , for example . This tells that has a much smaller value than , for example . The signum function defined as: For example , , . The floor function which gives the largest integer less than or equal to . For example , . The ceiling function which gives the smallest integer not less than . For example , . The square root of is a non-negative real number such that . Obviously always has to be non-negative. For example , while is not a proper expression since is negative. The -th root of is a real number such that . If is even needs to be non-negative, as is the case for the square root with . For example , , while is not a proper expression since the order of the root is even but is negative. If is an expression that depends on the value of , this evaluates the following sum For example if then the above sum becomes If is an expression that depends on the value of , this evaluates the sum over those indices for which the predicate becomes true. For example if and the predicate is : " is a prime and is less than " then the sum becomes Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting : "" this generalised sum becomes the previous easier sum If is an expression that depends on the value of , this evaluates the following product For example if then the above product becomes If is an expression that depends on the value of , this evaluates the product over those indices for which the predicate becomes true. For example if and the predicate is : " is a prime and is less than " then the product becomes Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting : "" this generalised product becomes the previous easier product The factorial function of argument defined through The binomial coefficient of the natural numbers and defined through