Determine whether each of the functions listed in parts 12 i

Determine whether each of the functions listed in parts 1-2 is or is not (a) injective and (b) surjective.

n-1 if n is odd

These are Abstract Algebra problems.

Solution

1.

i)

f(1)=1,f(0.9)=1

Hence , f is not injective

ii)

Let z be any integer

So, f(z)=z

Hence f is surjective

2.

i)

Let, f(n)=f(m)

Let m and n not equal

Case 1: m and n both odd

f(n)=n-1,f(m)=m-1

So, m=n which is contradiction

Case 2: m and n both even

f(n)=n+1,f(m)=m+1

giving,m=n

Case 3: m even ,n odd

f(m)=m+1,f(n)=n-1

m+1=n-1

m=n-2

which is contradiction as m and n are of opposite parity

Hence contradiciont in all cases

So,m=n

Hence, f is injetive

ii)\\

Let z be an integer

Not for an even integer we get an odd integer 1 larger than even number

For odd integer we get even integer 1 smaller than odd number

Case 1:z is even

f(z+1)=z

Case 2: z is odd

f(z-1)=z

Hence, f is surjective