## 1.2. Relations and Functions

### Examples 1.2.3:

Let A = {1, 2, 3, 4}, B = {14, 7, 234}, C = {a, b, c}, and R = real numbers.
Define the following relations:

*r*relates A and B via: 1 ~ 234, 2 ~ 7, 3 ~ 14, 4 ~ 234, 2 ~ 234*f*relates A and C via: {(1,c), (2,b), (3,a), (4,b)}*g*relates A and C via: {(1,a), (2,a), (3,a)}*h*relates R and itself via:*{(x,sin(x))}*

- The relation
*r*is not a function, because the element 2 from the set**A**is associated with two elements from**B**. - The relation
*f*is a function, because every element from**A**has exactly one relation from the set**C**. - The relation
*g*is not a function, because the element {4} from the domain**A**has no element associated with it. - The relation
*h*is a function with domain**R**, because every element*{x}*in**R**has exactly one element*{sin(x)}*associated with it.