In the case of each of the number-theoretic functions below, indicate whether th

Question

In the case of each of the number-theoretic functions below, indicate whether the function is a partial function or a total function. Simply write either the word “partial” or the word “total” on each line below, as appropriate. (Recall that a number-theoretic function has natural number arguments and returns a natural number as it value.)

            addition _______________________

            multiplication ______________________

            subtraction (so-called “minus”) ________________________

            monus ______________________

factorial function_______________________

             square root function _______________________

             squaring function __________________             

             polynomial function m = 2n + 3 __________________

             polynomial function m = 2n – 10 __________________

4. In the case of each of the number-theoretic functions below, indicate whether the function is an injective (or 1-to-1) function, a surjective (or “onto”) function, both, or neither. Simply write either the phrases “injective”, “surjective”, “both injective and surjective”, or “neither injective nor surjective” on each line below, as appropriate.

             addition _______________________

             multiplication ______________________

             subtraction (so-called “minus”) ________________________

             monus ______________________

factorial function_______________________ (Recall that 0! = 1! = 1.)

             squaring function __________________             

             polynomial function m = 2n + 3 __________________

polynomial function m = 2n (the “doubling function) __________________

the function defined by m = n + 1 if n is even and m = n – 1 if n is odd _________

Explanation / Answer

1)

addition ____TOTAL

            multiplication _______TOTAL

            subtraction (so-called “minus”) ________PARTIAL

            monus _________TOTAL

factorial function__________TOTAL

             square root function _______PARTIAL

squaring function ___________TOTAL   

             polynomial function m = 2n + 3 __________TOTAL

             polynomial function m = 2n – 10 ___________PARTIAL

2)

addition __________INJECTIVE

             multiplication ________INJECTIVE

             subtraction (so-called “minus”) ___________INJECTIVE

             monus ____________NEITHER

factorial function_____________NEITHER

             squaring function _____________INJECTIVE   

             polynomial function m = 2n + 3 __________INJECTIVE

polynomial function m = 2n (the “doubling function) _________INJECTIVE

the function defined by m = n + 1 if n is even and m = n – 1 if n is odd _________BOTH

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