Functions
2.0 Relation
2.0 Relation
Let $A$ and $B$ be two sets. Then a relation $R$ from set $A$ to set $B$ is a subset of $A \times B$.
Thus, $R$ is a relation from $A$ to $B$.
$$R \subseteq A \times B$$
For example, let
If $A$ and $B$ are non-void sets, $R$ is a relation from $A$ to $B$ and if $(a,b) \in R$, then, this relation is represented as $aRb$. This expression means, " $a$ is related to $b$ by the relation $R$ ".
Example 8. Which of the following are relation from $A$ to $B$, where $A = \{ a,e,i,o,u\} $ and $B = \{ p,q,r,s,t\} $
| i.${R_1} = \{ (a,p),(e,p),(o,s),(u,r)\} $ | iii.${R_3} = \{ (p,a),(p,e),(r,i)\} $ | v. ${R_5} = \{ (e,i),(a,o),(u,i)\} $ |
| ii.${R_2} = \{ (i,q),(e,t),(a,s),(p,r)\} $ | iv. ${R_4} = \{ (i,p),(i,q),(i,r),(i,t),(i,s)\} $ |
Solution:
| Relation | Classification |
| i.${R_1} = \{ (a,p),(e,p),(o,s),(u,r)\} $ | Here every element of ${R_1}$ belongs to $A \times B$, thus ${R_1} \subset A \times B$. Hence ${R_1}$ is a relation from $A$ to $B$. |
| ii.${R_2} = \{ (i,q),(e,t),(a,s),(p,r)\} $ | In the relation ${R_2}$ element $(p,r) \notin A \times B$, thus ${R_2} \not\subset A \times B$. Hence ${R_2}$ is not a relation from $A$ to $B$ |
| iii.${R_3} = \{ (p,a),(p,e),(r,i)\} $ | In the relation ${R_3}$ none of the elements belong to $A \times B$, thus ${R_3} \not\subset A \times B$. Hence ${R_3}$ is not a relation from $A$ to $B$ Here, the relation ${R_3}$ is actually a relation from $B$ to $A$. |
| iv. ${R_4} = \{ (i,p),(i,q),(i,r),(i,t),(i,s)\} $ | Here every element of ${R_4}$ belongs to $A \times B$, thus ${R_4} \subset A \times B$. Hence ${R_4}$ is a relation from $A$ to $B$. |
| v. ${R_5} = \{ (e,i),(a,o),(u,i)\} $ | In the relation ${R_5}$ none of the elements belong to $A \times B$, thus ${R_5} \not\subset A \times B$. Hence ${R_5}$ is not a relation from $A$ to $B$. In fact, the relation ${R_5}$ is a relation from $A$ to $A$. |
Example 9. Write down the relation between the two sets, $A = \{ 1,2,3,4\} $ and $B = \{ x:x < 10,\,x \in N\} $ as defined below in each case
| i. A relation $(a,b)$ from $A$ to $B$ such that $b = 2a$ | ii. A relation $(m,n)$ from $B$ to $B$ such that $n = m + 1$ | iii. A relation $(a,b)$ from $A$ to $B$ such that ${a^2} = b$ |
| iv. A relation $(p,q)$ from $B$ to $A$ such that every prime number in $B$ is related to every prime number in $A$ | v. A relation from $A$ to $A$ such that each and every element is related to the every element other than itself | |
Solution:
| Relation Defined | Relation |
| i. A relation $(a,b)$ from $A$ to $B$ such that $b = 2a$ | ${R_1} = \{ (1,2),(2,4),(3,6),(4,8)\} $ |
| ii. A relation $(m,n)$ from $B$ to $B$ such that $n = m + 1$ | ${R_2} = \{ (1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9)\} $ |
| iii. A relation $(a,b)$ from $A$ to $B$ such that ${a^2} = b$ | ${R_3} = \{ (1,1),(2,4),(3,9)\} $ |
| iv. A relation $(p,q)$ from $B$ to $A$ such that every prime number in $B$ is related to every prime number in $A$ | ${R_4} = \{ (2,2),(2,3),(3,2),(3,3),(5,2),(5,3),(7,2),(7,3)\} $ |
| v. A relation from $A$ to $A$ such that each and every element is related to the every element other than itself | ${R_5} = \{ (1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)\} $ |
