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I would like a more thorough understanding of how to determine the properties (reflexivity, symmetry, anti-symmetry, transitivity, completeness, asymmetry) of relations. I understand the idea in spoken words but have a hard time applying it mathematically. For example, the relation:

$$S \subset\mathbb{R^2}, \;\;xSy \iff y=\left|\left(\frac13\right)^x-1\right|$$

What steps should I take to go about determining the various properties of this relation? A step by step solution would be much appreciated.

1 Answers 1

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Using your relation: $$(x, y) \in S \subset\mathbb{R^2}, \;\;xSy \iff y=\left|\left(\frac13\right)^x-1\right|,\quad x, y \in \mathbb{R}$$ you need to determine if the following properties hold:

Reflexivity:
Is $x S x$ for all $x \in \mathbb{R}$? If so, reflexivity holds. If not, then reflexivity fails.

What can you say about whether or not $x = \left|\left(\frac13\right)^x-1\right|$ is true for all $x$ in $\mathbb{R}$?

Symmetry:
if $x S y$, is $y S x$ for all $x, y \in \mathbb{R}$?
If so, symmetry holds. If not, it fails.

Is it always the case that for $(x, y) \in \mathbb{R}^2$, if $y = \left|\left(\frac13\right)^x-1\right|$, then $x = \left|\left(\frac13\right)^y-1\right|$?
If not, then the relation $S$ is not symmetric.

Antisymmetry:
for all $x, y, z \in \mathbb{R}$, if $x S y$ and $y S x$, does this imply that $x = y$?
If so, the relation is antisymmetric. If not, the relation is not antisymmetric.

Is it always the case that if $y = \left|\left(\frac13\right)^x-1\right|$ and $x = \left|\left(\frac13\right)^y-1\right|$, then it follows that $x = y$?

Transitivity:
If $x S y$ and $ySz$, is $x S z$, for all $x, y, z \in \mathbb{R}$?
If so, then the relation is transitive; if not, then the relation is not transitive.

Is it always the case that if $y = \left|\left(\frac13\right)^x-1\right|$, and $\left|\left(\frac13\right)^y-1\right|$, then it follows that $z = \left|\left(\frac13\right)^x-1\right|$?

Unpack, in a similar manner, exactly what is required for a relation to satisfy the properties of completeness and asymmetry (i.e., use the definitions of a completeness and asymmetry, and test whether your relation $S$ meets the required conditions for those properties to hold.)

Note: if you can find any counterexamples to any given property, you thereby show that the a property doesn't hold for $S$, because a relation only has a property if it is holds for all elements of the set on which it is defined. Put differently, a property hold unless there is a counterexample that satisfies the "if(s)"..., but fails to satisfy the "then" part of the property's definition.


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    Thank you for the reply. So, if i wanted to express this mathematically (e.g. reflexivity) could I say that $\left(\frac13\right)^x-1 = x$ or $\left(\frac13\right)^x-1 = -x$, then $\left(\frac13\right)^x = x+1$, or $\left(\frac13\right)^x = 1-x$, which is clearly not true $\forall x \in \mathbb{R}$?2012-12-11
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    Yes, revok, that's correct!2012-12-11
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    Great! however I do have one more question. When determining e.g. symmetry, is it valid to do so by substitution? E.g. I solve for y and then plug it into x? This is how I determined symmetry: $(x=-x-1\land y=-x-1)\lor(x=y-1\land y=x-1)$ then $(x=x \land y=y)\lor (x=x-2\land y=y-2)$.. and then I compute the logic values meaning the relation is symmetric? I appreciate your help!2012-12-11
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    What if $x = 1, y = -2/3$, then $xSy$ Is it the case, then, that $ySx$? That is, is $1 = (1/3)^{-2/3} - 1$? If not, then we have $xSy$, but not $ySx$. Hence, $S$ is not symmetric.2012-12-11
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    Ok, so is the standard procedure to just plug in numbers and see what happens? I am looking for a systematic way to determine these properties. Is substitution valid at all?2012-12-11
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    More specifically, (as a partial solution) can I do this: $x=-y-1 \Rightarrow y=-x-1$ then $x=-(-x-1)-1 \Rightarrow y=-(-y-1)-1$?2012-12-11
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    Where do you get $x = -y -1$? A property holds for a relation UNLESS there is an example satisfying the "if" parts of the definitions above, but failing to satisfy the "then" parts of the definitions above. Reflexivity fails, symmetry fails, transitivity fails. I believe that antisymmetry holds (that is, for all those pairs of x, y, IF $x S y$ and $y S x$ THEN $x = y$. Since the only time that we have BOTH $x S y$ and $y S x$ is when $x = y = 0$, we have that indeed, $x = y$.2012-12-11
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    I understand that, and I think I haven't been clear with my question. I should have asked how can I FORMALLY prove these relations don't exist. That is what I was trying to do above with the $x=-y-1$ bit..2012-12-11
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    Thanks for all the insight. It's been very helpful!2012-12-11
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    You can FORMALLY prove that a property does not hold by providing an explicit counterexample. To disprove any statement that something holds for all p such that q, then r..., you need only show existence of a p such q, but not r. The one property that holds among the four I discuss is antisymmetry: try using logarithms to get the "x" out of the exponent.2012-12-11
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    You're welcome, revok. Feel free to edit your post (Use something like "EDIT: here's what I've got so far...". And send another comment; I'd be happy to look over your progress!2012-12-11
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    I appreciate that :)2012-12-11