Question

Determine whether f: Z x Z → Z is onto if..

a) f(m,n) = 2m-n

b) f(m,n) = m2-n2

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

d) f(m,n) = |m| - |n|

Answer 1

(a) Onto

(b) Not onto

(c) onto

(d) onto

Step-by-step explanation:

Let, A = Z × Z and B = Z

(a) f(m,n) = 2 m - n

An integer can be written as 2 times of another integer minus any third integer.

Thus, for every element of B there is a pre-image in A.

i.e, f is onto.

(b)
`f(m, n) = m^2 - n^2`

Since, there are lots of integers that can not obtain after subtracting the squares of two integers,

For eg : 1, 2, 4,..... etc

Thus, for these elements there is not any pre-image in A,

I.e. f is not onto.

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

An integer can be written as the sum of two integers and 1.

Thus, for every element of B there is a pre-image in A.

i.e, f is onto.

(d) f(m,n) = |m| - |n|

|x| = x if x > 0

While, |x| = -x if x < 0,

Thus, both |m| and |n| are always positive,

Every integer can be written as the difference of two positive integers.

Thus, for every element of B there is a pre-image in A.

i.e, f is onto.