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

f₃ = {(a, x), (b, x), (c, z), (d, z)}; A = {a, b, c, d}, B = {x, y, z}

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

So, is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all

⇔ f(a) = f(b)

⇒ a = b for all

Onto Function: – A function 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, is Surjection iff for each , there exists such that f(a) = b

Now, As given,

f₃ = {(a, x), (b, x), (c, z), (d, z)}

A = {a, b, c, d}, B = {x, y, z}

Thus we can clearly see that

Check for Injectivity:

Every element of A does not have different image from B

Since,

f₃(a) = x = f₃(b) and f₃(c) = z = f₃(d)

Therefore f is not One – One function

Check for Surjectivity:

Also each element of B is not image of any element of A

Hence f is not Onto.