Solve Discrete Math Problem: f(x,y)= 4x+y-4

[mamma_mia66]

I know I have to write an equation to solve the problem down. But I really don't know how to use the given information. I did it by enumeration, but I don't get it how this will be shown by an algebriac argument. Please some one help me at least with an idea.

If S = {1,2,3,4}, consider the function f:SxS-> N defined by f(x,y)= 4x+y-4. Determine the image of f, and show by an algebraic argument [not by enumeration] that f is one to one function on SxS. [hint: because S has only four elements, the difference of two of its elements is a multiple of 4 iff they are equal.]

 

[mathman]

f(x1,y1)-f(x2,y2)=4(x1-x2)+(y1-y2). This can be 0 if and only if x1=x2 and y1=y2.

4(x1-x2) is ALWAYS a multiple of 4, while (y1-y2) cannot be a multiple of 4 unless y1=y2.

 

[mamma_mia66]

Thank you so much. I get it now.