Answer to Question #345964 in Discrete Mathematics for asasas

Question

Determine whether each of these functions from Z to Z is one-to-one.

a. f(n) = n2

+ 1

b. f(n) = n

Answer 1

a. Let $n_1=-1, n_2=1, n_1, n_2\in \Z,n_1\not=n_2$

$$f(n_1)=f(-1)=(-1)^2+1=2$$

$$f(n_2)=f(1)=(1)^2+1=2$$

We see that

$$f(n_1)=f(-1)=1=f(1)=f(n_2),$$

but $n_1=-1\not=1=n_2.$

Therefore the function $f(n)=n^2+1, n\in \Z$ is not one-to-one from $\Z$ to $\Z.$

b. Let $f(n_1)=f(n_2).$ It means that

$$n_1=n_2$$

The function $f(n)=n$ is bijective (one-to-one ) from $\Z$ to $\Z.$