1. **Problem Statement:** Define relation and function. Give three examples of each by both ways (set and arrow diagram). Also, discuss 3 examples of relations that are not functions. Moreover, find the domain, co-domain, and range in each case. 2. **Definitions:** - A **relation** from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$, i.e., a set of ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. - A **function** is a relation where every element in the domain $A$ is related to exactly one element in the co-domain $B$. 3. **Important rules:** - In a function, no element in the domain maps to more than one element in the co-domain. - The domain is the set of all first elements in the ordered pairs. - The co-domain is the set $B$ to which elements are mapped. - The range is the set of all actual images of elements from the domain. --- ### Examples of Relations (not necessarily functions): **Example 1:** - Set form: $A=\{1,2,3\}$, $B=\{a,b\}$ - Relation $R=\{(1,a),(2,a),(3,b)\}$ - Domain: $\{1,2,3\}$ - Co-domain: $\{a,b\}$ - Range: $\{a,b\}$ - Arrow diagram: 1\rightarrow a, 2\rightarrow a, 3\rightarrow b **Example 2:** - $A=\{x,y\}$, $B=\{1,2,3\}$ - $R=\{(x,1),(x,2),(y,3)\}$ - Domain: $\{x,y\}$ - Co-domain: $\{1,2,3\}$ - Range: $\{1,2,3\}$ - Arrow diagram: x\rightarrow 1 and 2, y\rightarrow 3 **Example 3:** - $A=\{p,q\}$, $B=\{m,n\}$ - $R=\{(p,m),(q,m),(q,n)\}$ - Domain: $\{p,q\}$ - Co-domain: $\{m,n\}$ - Range: $\{m,n\}$ - Arrow diagram: p\rightarrow m, q\rightarrow m and n --- ### Examples of Functions: **Example 1:** - $A=\{1,2,3\}$, $B=\{a,b,c\}$ - Function $f=\{(1,a),(2,b),(3,c)\}$ - Domain: $\{1,2,3\}$ - Co-domain: $\{a,b,c\}$ - Range: $\{a,b,c\}$ - Arrow diagram: 1\rightarrow a, 2\rightarrow b, 3\rightarrow c **Example 2:** - $A=\{x,y,z\}$, $B=\{1,2\}$ - $f=\{(x,1),(y,2),(z,1)\}$ - Domain: $\{x,y,z\}$ - Co-domain: $\{1,2\}$ - Range: $\{1,2\}$ - Arrow diagram: x\rightarrow 1, y\rightarrow 2, z\rightarrow 1 **Example 3:** - $A=\{p,q\}$, $B=\{m,n\}$ - $f=\{(p,m),(q,n)\}$ - Domain: $\{p,q\}$ - Co-domain: $\{m,n\}$ - Range: $\{m,n\}$ - Arrow diagram: p\rightarrow m, q\rightarrow n --- ### Examples of Relations that are NOT Functions: **Example 1:** - $A=\{1,2\}$, $B=\{a,b\}$ - Relation $R=\{(1,a),(1,b),(2,a)\}$ - Here, 1 maps to both a and b, so not a function. **Example 2:** - $A=\{x,y\}$, $B=\{1,2\}$ - $R=\{(x,1),(x,2),(y,2)\}$ - x maps to 1 and 2, so not a function. **Example 3:** - $A=\{p\}$, $B=\{m,n\}$ - $R=\{(p,m),(p,n)\}$ - p maps to both m and n, so not a function. --- This completes the definitions, examples, and domain, co-domain, and range for each case.