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enter image description here

is this a relation? does every element in set $A$ need to have an image(one or many) in set b???

[ sorry question might be too simple for most of u guys here but just started learning about relations ]

Meet Lalwani
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    The image is a graphical representation of a relation, but the image itself needs to be interpreted in order to be considered a relation. A relation from $A$ to $B$ is any subset of $A\times B$, proper subset or otherwise. There is no requirement for every element of $A$ to appear as the first entry in one of the pairs in the relation. – JMoravitz Apr 21 '21 at 16:06
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    The pictured relation from $A={5,6,7,8,9,10}$ to $B={7,8,9,10,11,13}$ is the relation ${(5,7),(6,8),(7,9),(8,10),(9,11)}$ and does indeed satisfy the definition. – JMoravitz Apr 21 '21 at 16:07
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    "is this a relation?" Yes. Well, it's a picture of a relation. Elements of $A$ are being matched to elements of $B$. A relation is a collection of such matched pairs. "does every element in set A need to have an image(one or many) in set b?" No. Not every element of A needs to be matched from (indeed no elements need to be matched), not every element of B needs to be matched to. And elements may be matched to and from multiple, single, or no times.... Now a function is a special type of relation, where every element of $A$ must be matched once, so this is not a function. – fleablood Apr 21 '21 at 16:47

2 Answers2

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Suppose $A = \{1,2\}$ and $B = \{c,d\}$. As others commented, a relation is any subset of $A \times B$. So here are some relations:

$\emptyset $

$\{(1,c) \}$

$\{(1,c), (1,d) \}$

$\{ (1,d), (2,d)\}$

$\{ (1,c), (1,d), (2,c), (2,d) \}$ (everything in $A$ is related to everything in $B$)

$\{ (2,c)\}$

$\{ (2,c), (2,d)\}$

If you restrict things so $A = B$, then you get the opportunity to have elements related to themselves (reflexive), for the order not to matter (symmetric) and so forth. So

$\{ (1,1), (2,2)\}$ is a reflexive relation on $A \times A$. So is

$\{ (1,1), (2,2), (1,2)\}$.

For an example of symmetric, we could have

$\{ (1,2), (2,1)\}$

And so forth. You can make many examples.

RobertTheTutor
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Intuitively, a relation is a matching of some elements of $A$ to some elements of $B$.

Technically a relation is the the set of such matched pairs represented in a set of ordered pairs. If $a$ is matched to $b$ then $(a,b)$ is element of the relation $R$ which is the set of all such matched ordered pairs. And if $c$ is not matched to $d$ then the oredered pair $(c,d)$ is not an element of the set $R$.

And there is utterly no requirement, that every element of $A$ or that every element of $B$ gets matched. Indeed, it'd be possible that absolutely no elements are matched! That'd be the empty relation and represented by the empty set $\emptyset$.

I don't know if there is a term for a relation where every element of $A$ is matched. There could be. But then again that might not be an important concept. A function is a special type of relation where every element of $A$ is matched and it is matched to exactly one element. This image is not a function because not every element is matched. But that is not a requirement for relations in general.

Any type of pairing between sets, including the option to not pair, falls into the idea of a relation.

fleablood
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