(b) Construct a non-injective function from N× N to N. Justify your answer brief

Question

(b) Construct a non-injective function from N× N to N. Justify your answer briefly. If such a function does not exist, explain why Answer: There are infinitely many correct A trivial one: answers. Since f sends the entire set N × N to a single element of N, f is trivially non-injective. For a less trivial example, let: g:(x,y) x g is surjective, and every element of N is the image of infinitely many elements of N × N. Precisely, all pairs of the form: (a,y), for y = 0, 1, 2, . . . are mapped to a under f. Problem 8 Let:

Explanation / Answer

An injective function is a function in which for every f(x) there is exactly one x.

For example, lets say f(x) = 2x

Then for f(2) = 4, there exist exactly one x and that is 2

for f(2.5) = 5, there exist exactly one x and that is 2.5

Thus f(x) is injective function

Second example, lets say f(x) = x*x

Then for

f(2) = 4,

f(-2) = 4

there exist two x, ie x = 2, -2 for f(x) = 4

Thus this f(x) is not injective.

Now lets come to the original question. Here the function has two parameters and one output, everything as a natural number.

So lets out of the blue, we define

g(x, y) = x, such that x, y belong to Natural Numbers set (N)

Now this is clearly not an injective function,

Example, from definition of function

g(2, 6) = 2

g(2, 5) = 2

g(2, 4) = 2

Thus for g(x, y) = 2 we have (x,y) = (2, 6) or (x, y) = (2, 5) or (x, y) = (2, 4)

We dont have unique (x,y), this contradicts the definition of injective function.

Thus the function g(x,y) is not injective

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