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Let $A={a, b, c}$, and let $R=\{(a, b), (c, b), (a, b)\}$. Find the domain of $R$ and the image of $R$.

This would be very elementary, but I want to get my answer checked.

Let $R$ be a relation from $A$ to $B$. Then by definition, then the domain of the relation $R$ in symbols is $$\operatorname{Dom}(R)=\{a\in A\;\vert\;(a, b)\in R\text{ for some }b∈B\}$$

and the image of the relation $R$ in symbols is $$\operatorname{Im}(R)=\{b\in \;\vert\;(a, b)\in R\text{ for some }a\in A\}$$

So, $\operatorname{Dom}(R)=\{a, c\}, \operatorname{Im}(R)=\{b\}$.

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The answer in straightforward language:

$Dom(R)$=$\{a,c\}$ and $Image(R)$=$\{b\}$

Because in the $2$ distinct ordered pairs in $R$, the pre-image parts are $a$ and $c$ whereas the image is always $b$.

Your answers are thus correct.

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  • $\begingroup$ I first post an answer and then you incorporate that answer into your question and claim to check the validity of those!!! That's really unfair... $\endgroup$ Dec 29, 2015 at 15:53
  • $\begingroup$ I accidently pressed enter button, while I was writing, then I immediately edited it before reading this answer. $\endgroup$
    – buzzee
    Dec 29, 2015 at 15:53
  • $\begingroup$ Anyway, my answer is still valid however. $\endgroup$ Dec 29, 2015 at 15:54
  • $\begingroup$ By the way, my question is in the part of binary relation R, which has 2 ordered pairs as its elements, not 3 ordered pairs. $\endgroup$
    – buzzee
    Dec 29, 2015 at 15:59
  • $\begingroup$ @buzzee: for future reference, if you accidentally post too soon, you can always delete your post immediately after. It will not be permanently deleted, but it will become invisible to most users and no one will be able to comment post an answer (if the post is a question). Then you can edit to fix the post and undelete it afterwards. $\endgroup$
    – tomasz
    Dec 29, 2015 at 16:01

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