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.
- if and only if divides
- if and only if
- if and only if
- if and only if is a multiple of 5
- is the set of countries in Europe, if and only if and have a common border.
Symmetry can be easily checked by swapping and :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 is true, but is false.
iii. Symmetric. If , then
iv. Symmetric. If , A is an integer, then . In both cases, and are multiples of five.
v. Symmetric. If shares a border with , then shares a border with .
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: means that ? If so, then the relation is transitive.