Question

If $A=\{a,b,c,d\},B=\{p,q,r,s\}$, then which of the following are relations from $A$ to $B$?

• A. $R_1=\{(a,p),(b,r),(c,s)\}$
• B. $R_2=\{(q,b),(c,s),(d,r)\}$
• C. $R_3=\{(a,p),(a,q),(d,p),(c,r),(b,r)\}$
• D. $R_4=\{(a,p),(q,a),(b,s),(s,b)\}$

Answer 1

Answer: (A), (C)

$R_1=\{(a,p),(b,r),(c,s)\}$
$R_3=\{(a,p),(a,q),(d,p),(c,r),(b,r)\}$

From then definiation of Relation

Let $R$ be $a$ relation from $A$ to $B$

i.e., $R\subseteq A\times B,$ then

Domain of {$a:a\in A,(a,b)\in R$ for some $b\in B$}

Range of $R=${ $b:b\in B,(a,b)\in R$ for some $a\in A$}.

Therefore,

(A) $R,=\{(a,p),(b,s),(c,s)\}$ is a relation as $\{a,b,c\}\subseteq A$ and $\{p,r,s\}\subseteq B.$

(B) $R_2=\{(q,b),(c,s),(d,r)\}$ is not a relation as $q$ is not from set $A.$

(C) $R_3=\{(a,p),(a,q),(d,p),(c,r),(b,r)\}$ is a relation as $\{a,d,c,b\}\subseteq A$ and $\{p,q,r\}\subseteq B.$

(D)$R_4=\{(a,p),(q,a),(b,s),(s,b)\}$ is not a relation as $q$ and $s$ does not belong to set $A.$