Consider N as the set of all natural numbers
It is given that f: N → N: f(x) = x2 Ɐ x ∈ N
We know that
f(x1) = f(x2)
It can be written as
x12 = x22
So we get
x12 – x22 = 0
We get
(x1 – x2) (x1 + x2) = 0
Here, (x1 – x2) = 0 where x1 = x2
Hence, f is one-one.
Consider A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25}
Here, f: A → B: f(x) = x2
By substituting the values
f(1) = 12 = 1
f(2) = 22 = 4
f(3) = 32 = 9
f(4) = 42 = 16
Hence, every moment in A has unique image in B. Here, ∃ 25 ∈ B which has no pre-image in A f is into.