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What do I need to look for in order to tell if a relation on a set is a function?

Can somebody provide some advice?

For example, the relation is defined by $H$ on $A \times \mathcal P(A)$ for $a ∈ A$ and $B ⊆ A$, $a\, R\, B$ iff $a ∈ B$.

Where $A =\{0,1\}$ does this mean H is a function?

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    $\begingroup$ Check if the condition in the definition of Function is satisfied ... $\endgroup$ – Mauro ALLEGRANZA Nov 11 '14 at 15:34
  • $\begingroup$ Here $aRB$ should be changed into $aHB$. The $R$ is probably an abbreviation for 'relation' and comes from nowhere. $\endgroup$ – drhab Nov 11 '14 at 15:52
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A binary relation on a set $A$ is a collection of ordered pairs of elements of $A$. An element of $A$ can be related to many elements of $A$.

A function is a relation between a set $A$ and a set $B$ with the property that each element of $A$ is related to exactly one element of $B$.

Having said that, all functions are relations but not all relations are functions.

A relation on a set $A$ is a function iff every element of $A$ is related or mapped to only one element. If you find that an element of $A$ is related to more than one element then the given relation cannot be a function.

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  • $\begingroup$ Thank you but does this mean that A = {0,1} is a function? Im not sure how to justify it ? $\endgroup$ – RandomMath Nov 11 '14 at 15:42
  • $\begingroup$ @ASoni Set $A$ is not a function and is not under diagnose. You should check wether relation $H$ is a function. $\endgroup$ – drhab Nov 11 '14 at 15:49
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  • For every $a\in A$ can you find a $B\in\wp A$ satisfying $aHB$?
  • Is this $B$ unique in satisfying this condition?

If (and only if) the answer on both questions is 'yes' then $H$ is a function.

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  • $\begingroup$ So does that mean that because 0 ∈ A and it is also an element of P(A),due to that it means it is ∈B because B is a subset of A therefore A is a function? $\endgroup$ – RandomMath Nov 11 '14 at 15:48
  • $\begingroup$ $0$ is indeed an element of $A$ but not of $P(A)$. $\endgroup$ – drhab Nov 11 '14 at 15:51

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