Determine which formulas below define surjections from N * N to N and why?

a) f(a,b) = a+b


b) f(a,b) = ab


c) f(a,b) = ab(b+1)/2


d) f(a,b) = (a+1)b(b+1)/2


e) f(a,b) = ab(a+b)/2

Determine which formulas below define surjections from N * N to N and why?
N = {1,2,3,4,5,.....}





then...


a. .. this is not surjective... since a+b = 1 ... then no a∈N , b∈N that will satisfy the equation...








b. this is surjective... just consider b = 1... then a = n





c. i think this is also surjective... again... cannot verify strongly.. yet...





d. (a+1)b(b+1) /2 = 1


again ... no a,b ∈ N. not surjective.








e. ab(a+b)/2 = 5 ... no values.. not surjective.











§
Reply:If a,b are in N --%26gt; a+b, a·b are in N





Also a·b·(b+1)/2 or (a+1)·b·(b+1)/2 because one between b and (b+1) is even --%26gt; It's possible to divide by 2 (in N)








Also a·b·(a+b)/2 [ if a or b is even, evidently; if a and b are


odds --%26gt; a+b even]





All formulas a), b), c) d) and e) define surjections from


NxN to N





Saludos.