Answer to Question #123565 in Discrete Mathematics for nimra

Question

Let X= {1, 2, 3, 4}Y= {a, b, c} and Z={1,2}
1. Define K: Y→X as follows: K(a) = 1, K(b) = 2, K(c) = 3, and K(c) = 4. Is K onto? If not explain.
2. Using arrow diagram, find one function from X to Y that is Onto but not One-to-One
3. Define a function G: Y→Z that is onto and one-to-one and represent through matrix.

Answer 1

1) Given defined K is onto since every element of co-domain is in range of K. But defined K is not a function because one element has 2 images.

2) $1\to a, 2 \to b, 3 \to c, 4 \to c$ . This is function and not one-one since 3 and 4 has same image. This is onto also.

3) Function is not defined where function is one-one because co-domain has less element than domain. Hence, atleast two element have one image which can-not be one-one.