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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.

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1 Answer 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 $\forall x \in \mathbb{R}$? –  revok Dec 11 '12 at 13:03
    
Yes, revok, that's correct! –  amWhy Dec 11 '12 at 15:18
    
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! –  revok Dec 11 '12 at 15:45
    
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. –  amWhy Dec 11 '12 at 15:54
    
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? –  revok Dec 11 '12 at 16:00

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