help on relation

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February 2nd 2009, 05:57 AM
anon18

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help on relation

could anyone help with any part of this i think part 3 and 5 are symmetric im not sure about the rest
February 3rd 2009, 09:44 AM
stapel
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Which of the following relations R on sets X are reflexive, symmetric, transitive? Give proofs or counterexamples.

$X\, =\, \mathbb{N},\, (a,b)\,\in\, R$ if and only if $a$ divides $b$
$X\, =\, \mathbb{R},\, (a,b)\, \in\, R$ if and only if $a\, \leq \,b$
$X\, =\, \mathbb{C},\, (a,b)\, \in\, R$ if and only if $\left|a\right|\, =\, \left|b\right|$
$X\, =\, \mathbb{Z},\, (a,b)\, \in \, R$ if and only if $a\, -\, b$ is a multiple of 5
$X$ is the set of countries in Europe, $(a,b)\, \in \, R$ if and only if $a$ and $b$ have a common border.
February 3rd 2009, 10:13 AM
Referos

Quote:

i think part 3 and 5 are symmetric im not sure about the rest

Symmetry can be easily checked by swapping $a$ and $b$ :

i. Not symmetric. For instance, 1 divides 5, but 5 doesn't divide 1.
ii. Not symmetric. For instance $1 \leq 5$ is true, but $5 \leq 1$ is false.
iii. Symmetric. If $\left|a\right|\, =\, \left|b\right|$ , then $\left|b\right|\, =\, \left|a\right|$
iv. Symmetric. If $a - b = 5A$ , A is an integer, then $b - a = -5A$ . In both cases, $5A$ and $-5A$ are multiples of five.
v. Symmetric. If $a$ shares a border with $b$ , then $b$ shares a border with $a$ .
February 4th 2009, 03:11 AM
anon18

any idea which of the others are relexive or transitive?
February 4th 2009, 03:38 AM
Plato

Well, iii is an equivalence relation so it is all three: reflexive, symmetric and transitive.
For v, does any county share a border with itself? If so then that relation is reflexive.
Do that for each of the other relations: is it true that each term is related to itself?
Is this true: $a \le b\;\&\; b\le c$ means that $a \le c$ ? If so, then the relation is transitive.