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Question: If $X = \{1,2,3,4,5\}$ and $Y = \{1,3,5,7,9\}$, determine which of the following sets represent a relation? Options:

A) $R_1=\{(x,y):y=x+2,x\in X,y\in Y\}$;

B) $R_2=\{(1,1),(1,2),(3,3),(4,3),(5,5)\}$;

C) $R_3=\{(1,1),(1,3),(3,5),(3,7),(5,7),(7,3)\}$;

D) $R_4=\{(1,3),(2,5),(4,7),(5,9),(3,1)\}$;

I think the answer is both option A and D because $R_1 = \{(1,3),(3,5),(5,7)\}$. But when I see the solution it is given as options D. Am I missing something? Or is the answer provide wrong? Please elaborate so that I can understand.

Gary
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Sreeram M A
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  • Welcome to MSE. A question should be written in such a way that it can be understood even by someone who did not read the title. – José Carlos Santos Jun 29 '22 at 07:50
  • Oh OK I will edit it. – Sreeram M A Jun 29 '22 at 07:51
  • I would say that all but B) are relations, as they are subsets of $X \times Y$, but maybe I'm using a different definition to your book/teacher. What is the definition of "relation" that you're using? Is it possible that it meant "function" instead? – Theo Bendit Jun 29 '22 at 07:58
  • @TheoBendit the third option contains (7,3) too which I forgot to mention. So my answer that both option A and D are correct holds true right? – Sreeram M A Jun 29 '22 at 08:05
  • @SreeramMA In that case, I would agree. It's also possible that they meant to write for A), $R_1 = {(x, y) : y = x + 2, x \in X}$ (this is how I initially misinterpreted it, before I realised my error), which would make it not a relation between $X$ and $Y$. – Theo Bendit Jun 29 '22 at 08:08
  • @MauroALLEGRANZA Since $4 \notin Y$, $(2, 4) \notin R_1$ by definition. – Theo Bendit Jun 29 '22 at 08:08
  • @MauroALLEGRANZA is it necessary that if we state such a condition {(x,y): y = 2 + x, x∈X, y∈Y} then that it should hold true for every x∈X, then it would become a function right and not a relation. So (2,4) ∉ R1 in my opinion – Sreeram M A Jun 29 '22 at 08:11

2 Answers2

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You're quite right in that $$ R_1 = \{(x,y):y=x+2,\ x \in X,\ y \in Y\} = \{(1,3),(3,5),(5,7)\} $$ which is a relation over $X$ and $Y$. You can also tell it's a relation because the "$x \in X, y \in Y$" part of its definition implies $R_1 \subseteq X \times Y$. I'm guessing the author intended the first set to be $$ \{(x,x+2):x \in X\} $$ which would not be a relation over $X$ and $Y$. (Or maybe the author meant to say "function" instead of "relation", as in Theo Bendit's comment.)

We know $R_2$ is not a relation since $(1,2) \not\in X \times Y$. Both $R_3$ and $R_4$ are relations, and you can check the first coordinates all belong to $X$ and second coordinates all belong to $Y$.

Rebecca J. Stones
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As Theo Bendt says in his comment, a relation between sets $X$ and $Y$ is simply a subset of $X \times Y$, that is, if you choose $R \subseteq X \times Y$, then you can define a relation, let's call it $\lt_R$, by stating that $x \lt_R y$ for $x \in X, y \in Y$ if and only if $(x,y) \in X \times Y$.

Thus, $R_1$ does not define a relation, since $R_1=\{(1,3),(2,4),(3,5),(4,6),(5,7)\}$, which is not a subset of $X \times Y$, as $4 \notin Y$.

Applying the same analysis shows that $R_2$ also doesn't define a relation, but $R_3$ and $R_4$ both do.

j4nd3r53n
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  • My doubt is why should (2,4) and (4,6) belong to R1. So if we have written the relation in set builder form the relation should hold true for every value of x? Pls answer – Sreeram M A Jun 29 '22 at 08:19
  • $R_1$ is constructed by taken elements $x \in X$ as the first component and $x+2$ as the second: $(x,x+2)$ - this means $R_1$ is not a subset of $X \times Y$, because the pair $(2,4)$ is in $R_1$, and $4 \notin Y$. – j4nd3r53n Jun 29 '22 at 08:25
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    but the condition y∈Y is there so (2, 4) ∉ R1 since y∉Y, so it doesn't matter right – Sreeram M A Jun 29 '22 at 09:00
  • For $R_1$ to be a relation, it is true that $y$ must be in $Y$, but the problem starts at the other end, so to speak: we construct $R_1$ as a set of pairs of numbers, which is valid; unfortunately it doesn't define a relation between $X$ and $Y$. It is also true that a subset of $R_1$ does define a relation between $X$ and $Y$ - a relation does not have to define a relationship between all pairs of elements in the two sets. However, $R_1$ as it is defined, does not define a relation. – j4nd3r53n Jun 29 '22 at 09:07