In each case below determine whether the function given is i

In each case below, determine whether the function given is injective (one-to-one) and prove your answer. (a) f: Z rightarrow R where f(x) = (3x - 4)/8. (b) f: N times N rightarrow Q where f(a, b) = a/b. (c) f: N rightarrow R where f(n) = 1/n. (d) f: R rightarrow Z where f(x) = [x].

Solution

a)

Let, f(x)=f(y)

(3x-4)/8=(3y-4)/8

3x-4=3y-4

3x=3y

x=y

So, f is injective

b)

Not injective

BEcause,

f(a,b)=f(ma,mb) =(ma)/(mb)=a/b for any natural number m

c)

Let, f(n)=f(m)

1/n=1/m

n=m

So, f is injective

d)

Not injective

example:

f(1)=f(1.1)=1