Question

Which of the following functions from A to B are one – one and onto?

f₁ = {(1, 3), (2, 5), (3, 7)}; A = {1, 2, 3}, B = {3, 5, 7}

Answer 1

One – One Function: – A function f: A → B is said to be a one – one functions or an injection if different elements of A have different images in B.

So, f: A → B is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all a, b ∈ A

⇔ f(a) = f(b)

⇒ a = b for all a, b ∈ A

Onto Function: – A function f: A → B is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, f: A → B is Surjection iff for each b ∈ B, there exists a ∈ B such that f(a) = b

Now, As given,

f₁ = {(1, 3), (2, 5), (3, 7)}

A = {1, 2, 3}, B = {3, 5, 7}

Thus we can see that,

Check for Injectivity:

Every element of A has a different image from B

Hence f is a One – One function

Check for Surjectivity:

Also, each element of B is an image of some element of A

Hence f is Onto.