# help on relation

• Feb 2nd 2009, 05:57 AM
anon18
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
• Feb 3rd 2009, 09:44 AM
stapel
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Quote:

Which of the following relations R on sets X are reflexive, symmetric, transitive? Give proofs or counterexamples.

1. $X\, =\, \mathbb{N},\, (a,b)\,\in\, R$ if and only if $a$ divides $b$
2. $X\, =\, \mathbb{R},\, (a,b)\, \in\, R$ if and only if $a\, \leq \,b$
3. $X\, =\, \mathbb{C},\, (a,b)\, \in\, R$ if and only if $\left|a\right|\, =\, \left|b\right|$
4. $X\, =\, \mathbb{Z},\, (a,b)\, \in \, R$ if and only if $a\, -\, b$ is a multiple of 5
5. $X$ is the set of countries in Europe, $(a,b)\, \in \, R$ if and only if $a$ and $b$ have a common border.

• Feb 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
$a$.

• Feb 4th 2009, 03:11 AM
anon18
any idea which of the others are relexive or transitive?
• Feb 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.