could anyone help with any part of this i think part 3 and 5 are symmetric im not sure about the rest
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Which of the following relations R on sets X are reflexive, symmetric, transitive? Give proofs or counterexamples.
- $\displaystyle X\, =\, \mathbb{N},\, (a,b)\,\in\, R$ if and only if $\displaystyle a$ divides $\displaystyle b$
- $\displaystyle X\, =\, \mathbb{R},\, (a,b)\, \in\, R$ if and only if $\displaystyle a\, \leq \,b$
- $\displaystyle X\, =\, \mathbb{C},\, (a,b)\, \in\, R$ if and only if $\displaystyle \left|a\right|\, =\, \left|b\right|$
- $\displaystyle X\, =\, \mathbb{Z},\, (a,b)\, \in \, R$ if and only if $\displaystyle a\, -\, b$ is a multiple of 5
- $\displaystyle X$ is the set of countries in Europe, $\displaystyle (a,b)\, \in \, R$ if and only if $\displaystyle a$ and $\displaystyle b$ have a common border.
Symmetry can be easily checked by swapping $\displaystyle a$ and $\displaystyle b$:i think part 3 and 5 are symmetric im not sure about the rest
i. Not symmetric. For instance, 1 divides 5, but 5 doesn't divide 1.
ii. Not symmetric. For instance $\displaystyle 1 \leq 5 $ is true, but $\displaystyle 5 \leq 1$ is false.
iii. Symmetric. If $\displaystyle \left|a\right|\, =\, \left|b\right|$, then $\displaystyle \left|b\right|\, =\, \left|a\right|$
iv. Symmetric. If $\displaystyle a - b = 5A$, A is an integer, then $\displaystyle b - a = -5A$. In both cases, $\displaystyle 5A$ and $\displaystyle -5A$ are multiples of five.
v. Symmetric. If $\displaystyle a$ shares a border with $\displaystyle b$, then $\displaystyle b$ shares a border with $\displaystyle a$.
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: $\displaystyle a \le b\;\&\; b\le c$ means that $\displaystyle a \le c$? If so, then the relation is transitive.