1.

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}

Answer»

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,

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.


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